# -*- coding: utf-8 -*-
"""Archive Labeling Sequences: Code

Automatically generated by Colaboratory.

Original file is located at
    https://colab.research.google.com/drive/1pGfgQWvJR1IAG3t4dNnrTnc07UvyV4xC

## Counting Occurrences Slowly

The plt/mz scheme program for [A163500](https://oeis.org/A163500):

```
(define (count-matches re str start-pos)
  (let ((m (regexp-match-positions re str start-pos)))
    (if m (+ 1 (count-matches re str (+ (caar m) 1))) 0)))

(define (matches-n-in-zero-to-k fn n)
  (do ((sum-so-far 1)
       (k (+ n 1))
       (re (regexp (format "~a" n))))
    ((fn sum-so-far k) k)
    (when (equal? 0 (modulo k 1000000)) ;; this is just a progress indicator
      (display (format "~a ~a ~a\n" n k sum-so-far)))
    (set! k (+ k 1))
    (set! sum-so-far
      (+ sum-so-far (count-matches re (format "~a" k) 0)))))

(define (s f n)
  (display (matches-n-in-zero-to-k f n)))
;; where f should be one of = or > depending on which sequence you care about.
;; this could be made much more efficient, of course. In particular, the
;; initial sequences up to the first x of m digits have serious regularity.
```

This is a very straightforward, if slow, way to compute the interesting values, and here is a python (mostly-)equivalent:
"""

import re

def count_matches(rexp, str):
  return len(re.findall(rexp, str))

from tqdm import tqdm

def matches_n_in_zero_to_k(fn, n):
  sum_so_far = 1
  k = n + 1
  rexp = re.compile(str(n))
  for i in tqdm(range(int(10e10+1))):
    if fn(sum_so_far, k):
      return k
    k += 1
    sum_so_far += count_matches(rexp, str(k))

if not "SKIP":
  for d in (1, 2, 3, 4, 5, 6, 7, 8, 9):
    print(f"{d=}: {matches_n_in_zero_to_k(lambda a, b: a==b, d)}")

"""This is expected to produce something beginning with:
```
  0%|          | 199979/100000000001 [00:00<48:36:28, 571466.44it/s]
d=1: 199981
  0%|          | 28263824/100000000001 [00:53<52:54:32, 524862.05it/s]
d=2: 28263827
  0%|          | 371599979/100000000001 [12:03<53:51:24, 513853.84it/s]
d=3: 371599983
  0%|          | 499999979/100000000001 [16:14<53:53:35, 512844.95it/s]
d=4: 499999984
  2%|▏         | 2159642255/100000000001 [1:07:10<50:43:35, 535772.12it/s]
```
perhaps with different actual and estimated times from tqdm.

But as shown in the last line, it may take a couple of days to complete a search for d=5, on free Google cloud compute in 2023. Of course computer time is often less valuable than person-time?

The same program applies for [A164321](https://oeis.org/A164321),
we are simply now passing greater-than to the matcher:
"""

if not "SKIP":
  for d in (1, 2, 3, 4, 5, 6, 7, 8, 9):
    print(f"{d=}: {matches_n_in_zero_to_k(lambda a, b: a > b, d)}")

"""This proceeds similarly:
```
  0%|          | 199989/100000000001 [00:00<52:11:58, 532144.21it/s]
d=1: 199991
  0%|          | 28263825/100000000001 [00:55<54:26:39, 510062.37it/s]
d=2: 28263828
  0%|          | 371599989/100000000001 [12:45<56:59:18, 485615.58it/s]
d=3: 371599993
  0%|          | 499999989/100000000001 [16:04<53:18:11, 518522.18it/s]
d=4: 499999994
  0%|          | 18095952/100000000001 [00:33<51:40:23, 537468.56it/s]
```

## Counting in Any Base

Generalizing this to any base:
"""

def strbase(n, b):
  """Convert a python numeric value n to a base b list of numeric values.
  E.g. convert 23 to [1, 11] for base b=12.
  """
  return [n] if n < b else strbase(n // b, b) + [n % b]

# The above reaches a recursion limit and crashes for long enough digit strings.
def strbase(n, b):
  lst = []
  while n >= b:
    lst.append(n % b)
    n = n // b
  lst.append(n)
  return list(reversed(lst))

import sys
sys.set_int_max_str_digits(0) # 4300 is default. 0 disables.
# The intent is against denial of service attacks, but in this case we may
# actually care about a great many digits.

def unstrbase(digits, base=10):
  """Convert a base b list of numeric values to a single numeric value.
  E.g., convert [1, 11] in base=12 to 1*12^1 + 11*12^0 = 23
  """
  if not "debug": print(f"unstrbase({digits}, {base})")
  sum = 0
  carry = 0
  for place in range(len(digits)):
    value = digits[-(place+1)] + carry
    carry = 0
    if value >= base:
      carry = value // base
      value = value % base
    if not "debug": print(f"{sum=} += {value} * {base} ** {place}")
    sum += value * base ** place
  return sum + carry * base ** len(digits)

def slow_count_matches_to_k(target, limit, base=10):
  assert target < base
  count = 0
  for i in range(1, limit+1):
    digits = strbase(i, base)
    count += sum([1 for d in digits if d == target])
  return count

assert 199981 == slow_count_matches_to_k(1, 199981)

"""## Counting Quickly, by Place

As a prelude to speeding things up, let's count the contribution of each digit:
"""

from collections import defaultdict
def slow_count_digits_by_place(target, limit, base=10):
  counts = defaultdict(int)
  counts[1]=0
  for x in range(1, limit+1):
    digs = strbase(x, base)
    for i, digit in enumerate(digs):
      place = len(digs) - i
      if digit == target:
        counts[place] += 1
  return list(map(lambda a: a[1], sorted(counts.items(), reverse=True)))

expected = [99982, 20000, 20000, 20000, 20000, 19999]
assert sum(expected) == 199981
assert expected == slow_count_digits_by_place(1, 199981)

"""Now we can use a closed form for the contribution to the count at each place, and thus speed things up."""

def count_nonzero_at_place(digit_to_count, limit, this_digit, this_place, base=10, debug=False):
  assert digit_to_count > 0
  lpv = base**this_place      # Larger of two relevant place values
  spv = base**(this_place-1)  # Smaller of two relevant place values
  until_target_digit = (limit // lpv) * spv
  if this_digit < digit_to_count:
    return until_target_digit
  elif this_digit > digit_to_count:
    return until_target_digit + spv
  else:
    return until_target_digit + (limit % spv) + 1

def count_zero_at_place(digit_to_count, limit, this_digit, this_place, base=10, debug=False):
  assert digit_to_count == 0
  lpv = base**this_place      # Larger of two relevant place values
  spv = base**(this_place-1)  # Smaller of two relevant place values
  if this_digit == 0:
    return (limit // lpv - 1) * spv + limit % spv + 1
  else:
    return (limit // lpv) * spv

def counts_at_each_place(digit_to_count, limit, base=10, debug=False):
  if debug: print("\n")
  limit_digits = strbase(limit, base)
  length = len(limit_digits)
  counter = count_nonzero_at_place if digit_to_count > 0 else count_zero_at_place
  return [counter(digit_to_count, limit, this_digit, length - i, base=base, debug=debug)
          for i, this_digit in enumerate(limit_digits)]

def count_by_digits(digit_to_count, limit, base=10, debug=False):
  counts_by_digit = counts_at_each_place(digit_to_count, limit, base=base, debug=debug)
  if debug:
    print(f"{counts_by_digit=}")
  return sum(counts_by_digit)

"""Time to check that (a random sample of) the per-digit counts match the slow counts for each base and target:"""

from tqdm import tqdm
import random
for base in range(2, 22):
  num_digits_to_check = 4  # We will check a little more, too.
  for x in tqdm(range(base**num_digits_to_check + base*num_digits_to_check), desc=f"%{base}"):
    for target in range(base):
      if random.random() < 1 / (2**base):
        slow = slow_count_digits_by_place(target, x, base=base)
        fast = count_by_digits(target, x, base=base)
        assert sum(slow) == fast, f"{target}@{x=}{strbase(x, base)}(%{base}): {slow=}={sum(slow)} {count_by_digits(target, x, base=base, debug=True)=}"

a1 = 199981
assert a1 == count_by_digits(1, a1)

"""## Verifying Table 1: Unbounded Binary Search

"""

Table_1 = [(1, 199981, 199991),
           (2, 28263827, 28263828),
           (3, 371599983, 371599993),
           (4, 499999984, 499999994),
           (5, 10000000000, 5555555555),
           (6, 9500000000, 6666666666),
           (7, 9465000000, 7777777777),
           (8, 9465000000, 8888888888),
           (9, 10000000000, 9999999999),
           ]

import time
from tqdm import tqdm
def unbounded_binary_ge_search(digit=1, base=10, safeleft=2, width=1, debug=False, tqdm_after=15):
  fn = lambda x: count_by_digits(digit, x, base=base)
  eq, gt = None, None
  limit = base**base if digit > 0 else base**(2*base)
  starttime = time.time()
  bar = None
  while safeleft <= limit and (gt is None or eq is None):
    upper = safeleft + width
    fnsl = fn(safeleft)
    if bar is None and time.time() - starttime > tqdm_after:
      bar = tqdm(total=limit, desc=f"{digit=} {base=}")
    if bar:
      bar.update(safeleft - bar.n)
    if eq is None and width == 1 and fn(upper) == upper:
      eq = upper
      safeleft, width = upper, 1
      continue
    if gt is None and width == 1 and fn(upper) > upper:
      gt = upper
      safeleft, width = upper, 1
      continue
    if fnsl > safeleft + 1:
      # If fn(safeleft) is larger than safeleft, then even if none of the values
      # between safeleft and fn(safeleft) - 1 have the digit of interest,
      # fn(safeleft) will still remain larger, so we can skip ahead a bit.
      safeleft, width = fnsl - 1, 1
    elif width == 1 or fn(upper) < safeleft:
      safeleft, width = upper,    width * 2
    else:
      safeleft, width = safeleft, width // 2
  return gt, eq

for d, aeq, agt in Table_1:
  gt, eq = unbounded_binary_ge_search(digit=d, base=10)
  assert eq == aeq, f"{d=} {aeq=} {eq=}"
  assert gt == agt, f"{d=} {agt=} {gt=}"
  print(f"({d}, {eq}, {gt}),")

"""## Zeroes"""

a0 = 100559404366
assert a0 + 1 == count_by_digits(0, a0)
assert 120987654321 == count_by_digits(0, 111111111111)
zgt, zeq = unbounded_binary_ge_search(0, base=10)
assert zeq is None, zeq
assert a0 == zgt, zgt

slow_count_matches_to_k(0, 32, 2)

"""## Multiple Overlapping Digits

### Counting slowly
"""

# Note that this way of counting double-counts stickers by place.
# We like it because it's compatible with single-digit counting; the formula for this applies to single digits also.
assert 2 == count_matches("(?=11)", "1113")
# So counting this way, we're saying the sticker 11 *could be* in two different
# places in 1113, even though an actual "11" sticker would only be in one of them.

from tqdm import tqdm
import re
def multi_matches_n_in_zero_to_k(stop_criterion, digitstring, limit=int(10e10)):
  sum_so_far = 0
  iter = range(1, limit+1)
  if limit > 10e5:
    iter = tqdm(iter)
  rexp = re.compile("(?=" + str(digitstring) + ")")
  for i in iter:
    if stop_criterion(sum_so_far, i):
      return sum_so_far, i
    sum_so_far += count_matches(rexp, str(i))
  return sum_so_far, i

assert (40, 1113) == multi_matches_n_in_zero_to_k(lambda a, b: a>=b, 11, limit=1113)

def multi_matches_by_place(digitstring, limit):
  digitstring = str(digitstring)
  pattern_len = len(digitstring)
  sum_so_far = [0 for i in str(limit)]
  iter = range(1, limit+1)
  if limit > 10e5:
    iter = tqdm(iter)
  for i in iter:
    as_str = str(i)
    for place in range(len(as_str)):
      extract = as_str[-(place + pattern_len) : -place] if place > 0 else as_str[-(place + pattern_len):]
      if digitstring == extract:
        sum_so_far[place] += 1
      #print(f"{i=} [{-(place+pattern_len)}:{-place}] = [{extract}]=?{digitstring} : {sum_so_far}")
  return sum_so_far

assert [12, 14, 14, 0] == multi_matches_by_place(11, 1113)
# Note: the 12 is for the right-most place, that is to say the last two digits.
# The 14s are for the middle two digits and the left two digits.
# The last zero is always zero for multi-digit values.

def slow_multi_matches_by_base(digits_as_list, limit, base=12):
  n_found = 0
  for value in range(1, limit+1):
    value_as_list = strbase(value, base)
    n_intervals = len(value_as_list) - len(digits_as_list) + 1
    for i in range(n_intervals):
      #print(f"{value_as_list}[{i}]")
      if all(value_as_list[i+j] == digits_as_list[j]
             for j in range(len(digits_as_list))):
        n_found += 1
  return n_found

# single-digit sanity
slow = slow_multi_matches_by_base([0], 3, base=2)
# 1, 10, 11
assert 1 == slow, slow

slow = slow_multi_matches_by_base([1], 3, base=2)
# 1, 10, 11
assert 4 == slow, slow

slow = slow_multi_matches_by_base([2], 8, base=3)
# 1, 2, 10, 11, 12, 20, 21, 22
assert 6 == slow, slow

# 10 in binary up to 15=1111: 1, 10*, 11, 100*, 101*, 110*, 111, 1000*, 1001*, 1010**, 1011*, 1100*, 1101*, 1110*, 1111 = 12
slow = slow_multi_matches_by_base([1, 0], 15, base=2)
assert 12 == slow, slow

# 11 in ternary up to 80=2222:
#   1  ,    2 ,   10  ,   11*,   12,   20 ,   21  ,   22 ,  100  ,  101,
# 102  ,  110*,  111**,  112*,  120,  121 ,  122  ,  200 ,  201  ,  202,
# 210  ,  211*,  212  ,  220 ,  221,  222 , 1000  , 1001 , 1002  , 1010,
#1011* , 1012 , 1020  , 1021 , 1022, 1100*, 1101* , 1102*, 1110**, 1111***,
#1112**, 1120*, 1121* , 1122*, 1200, 1201 , 1202  , 1210 , 1211* , 1212,
#1220  , 1221 , 1222  , 2000 , 2001, 2002 , 2010  , 2011*, 2012  , 2020,
#2021  , 2022 , 2100  , 2101 , 2102, 2110*, 2111**, 2112*, 2120  , 2121,
#2122  , 2200 , 2201  , 2202 , 2210, 2211*, 2212  , 2220 , 2221  , 2222
slow = slow_multi_matches_by_base([1, 1], 80, base=3)
assert 27 == slow, slow

# 12 in ternary up to 80=2222:
#   1 ,    2 ,   10 ,   11 ,   12*,   20 ,   21 ,   22 ,  100 ,  101 ,
# 102 ,  110 ,  111 ,  112*,  120*,  121*,  122*,  200 ,  201 ,  202 ,
# 210 ,  211 ,  212*,  220 ,  221 ,  222 , 1000 , 1001 , 1002 , 1010 ,
#1011 , 1012*, 1020 , 1021 , 1022 , 1100 , 1101 , 1102 , 1110 , 1111 ,
#1112*, 1120*, 1121*, 1122*, 1200*, 1201*, 1202*, 1210*, 1211*, 1212**,
#1220*, 1221*, 1222*, 2000 , 2001 , 2002 , 2010 , 2011 , 2012*, 2020 ,
#2021 , 2022 , 2100 , 2101 , 2102 , 2110 , 2111 , 2112*, 2120*, 2121*,
#2122*, 2200 , 2201 , 2202 , 2210 , 2211 , 2212*, 2220 , 2221 , 2222
slow = slow_multi_matches_by_base([1, 2], 80, base=3)
assert 27 == slow, slow

"""### Counting quickly by place"""

def fast_multi_matches_by_place(digit_list, limit, base=10):
  if isinstance(digit_list, int):
    digit_list = strbase(digit_list, base)
  digits_value = unstrbase(digit_list, base)
  if 0 == digits_value:
    return counts_at_each_place(0, limit, base=base)
  pattern_len = len(digit_list)
  limit_as_digits = strbase(limit, base)
  sum_so_far = [0 for i in limit_as_digits]
  for place in range(len(limit_as_digits)):
    extract = limit_as_digits[-(place + pattern_len) : -place] if place > 0 else limit_as_digits[-(place + pattern_len):]
    extract_val = unstrbase(extract, base)
    sum_so_far[place] = (limit // base**(place + pattern_len)) * base**place
    if extract_val < digits_value:
      pass
    elif extract_val == digits_value:
      sum_so_far[place] += (limit % base**(place)) + 1
    else:
      sum_so_far[place] += base**(place)
  return sum_so_far

for dstr in (0, 1, 2, 10, 11, 15, 22, 23, 32, 99, 100, 101, 999, 1000, 1001, 1234):
  for limit in (10, 100, 1000, 2000, 3245, 103298): #, 2345678):
    slow_count, _ = multi_matches_n_in_zero_to_k(lambda a, b: False, dstr, limit)
    by_place = multi_matches_by_place(dstr, limit)
    assert slow_count == sum(by_place), f"{dstr=} {limit=} {slow_count=}, {by_place=}, {sum(by_place)=}"
    fast_count = fast_multi_matches_by_place(dstr, limit)
    assert sum(by_place) == sum(fast_count), f"{dstr=} {limit=} {slow_count=} {by_place=} {fast_count=}"

from tqdm import tqdm
import random
for base in range(2, 22):
  num_digits_to_check = 4  # We will check a little more, too.
  limit = base**num_digits_to_check + base*num_digits_to_check
  for x in tqdm(range(limit), desc=f"%{base}"):
    if base > 4 and random.random() > 2 / limit: continue
    for target in range(base**3):
      if base > 4 and random.random() > 2 / base**3: continue
      slow = slow_multi_matches_by_base(strbase(target, base), x, base=base)
      fast = fast_multi_matches_by_place(target, x, base=base)
      assert slow == sum(fast), f"{target}@{x=}{strbase(x, base)}(%{base}): {slow=} {fast_multi_matches_by_place([target], x, base=base)=}"

"""### Searching for eq and gt"""

from tqdm import tqdm
import time
def multidigit_unbounded_eq_search(digits=10, base=10, safeleft=2, width=1, sizebound=0, timebound=None, debug=False):
  starttime = time.time()
  fn = lambda x: sum(fast_multi_matches_by_place(digits, x, base=base))
  gt = None
  bar = None
  if sizebound:
    bar = tqdm(total=sizebound)
  if timebound:
    bar = tqdm(total=timebound)
  size = 0
  while True:
    upper = safeleft + width
    if debug: print(f"{digits} [{safeleft} -- {upper}]: {fn(upper)=}")
    fnsl = fn(safeleft)
    if sizebound:
      size = len(str(safeleft))
      bar.update(size - bar.n)
      if size > sizebound:
        return -safeleft, gt
    if timebound:
      elapsed = time.time() - starttime
      bar.update(elapsed - bar.n)
      if timebound < elapsed:
        return -safeleft, gt
    if not gt and fn(upper) > upper:
      gt = upper
    if width == 1 and fn(upper) == upper:
      if not gt:
        for gtsearch in range(2 * 10**len(str(digits))):
          test = upper + gtsearch
          if fn(test) > test:
            gt = test
            break
      return upper, gt
    if fnsl > safeleft + 1:
      # If fn(safeleft) is larger than safeleft, then even if none of the values
      # between safeleft and fn(safeleft) - 1 have the digits of interest,
      # fn(safeleft) will still remain larger, so we can skip ahead a bit.
      safeleft, width = fnsl - 1, 1
    elif width == 1 or fn(upper) < safeleft:
      safeleft, width = upper,    width * 2
    else:
      safeleft, width = safeleft, width // 2

import time
from tqdm import tqdm
def multidigit_all_eq_search(digits=10, base=10, safeleft=2, width=1,
                             first_and_last=False, sizebound=0, timebound=None,
                             debug=False):
  starttime = time.time()
  fn = lambda x: sum(fast_multi_matches_by_place(digits, x, base=base))
  gt = None
  bar = None
  n_found = 0
  if sizebound and sizebound > 100:
    bar = tqdm(total=sizebound)
  if timebound and timebound > 10:
    bar = tqdm(total=timebound)
  size = 0
  all_eqs = []
  while True:
    upper = safeleft + width
    if debug: print(f"{digits} [{safeleft} -- {upper}]: {fn(upper)=}")
    fnsl = fn(safeleft)
    if sizebound:
      size = len(str(safeleft))
      if bar: bar.update(size - bar.n)
      if size > sizebound:
        return all_eqs, n_found, safeleft
    if timebound:
      elapsed = time.time() - starttime
      if bar: bar.update(elapsed - bar.n)
      if timebound < elapsed:
        return all_eqs, n_found, safeleft
    if width == 1 and fn(upper) == upper:
      n_found += 1
      if first_and_last and len(all_eqs)>1:
        all_eqs[-1] = upper
      else:
        all_eqs.append(upper)
      #print(upper)
      safeleft, width = upper, width
    if fnsl > safeleft + 1:
      # If fn(safeleft) is larger than safeleft, then even if none of the values
      # between safeleft and fn(safeleft) - 1 have the digits of interest,
      # fn(safeleft) will still remain larger, so we can skip ahead a bit.
      safeleft, width = fnsl - 1, 1
    elif width == 1 or fn(upper) < safeleft:
      safeleft, width = upper,    width * 2
    else:
      safeleft, width = safeleft, width // 2

if not "SKIP":
  ten_all_eq, n_found, final_bound = multidigit_all_eq_search(10, timebound=30)
  max(ten_all_eq), final_bound

if not "SKIP":
  for target, base in [(2, 2), (4, 2), (8, 2), (16, 2), (32, 2), (64, 2), (128, 2), (256, 2), (3, 3), (9, 3)]:
    all_eq, n_found, final_bound = multidigit_all_eq_search(target, base=base, sizebound=250)
    print(f"({target=}, {base=}, {strbase(target, base)=}, {n_found=}, {all_eq=}),")

"""### Multi-digit table with (dd, a_=, a_>)"""

# In case you don't have hours of compute, here are the terms we've found already:
# Two-digit values take 2-3s on Google Colab free tier. Three-digit take about 45m-1h each.
multidigit_table = [
  (1, 199981, 199991),
  (2, 28263827, 29705217),
  (3, 371599983, 380847105),
  (4, 499999984, 499999994),
  (5, 10000000000, 5592858625),
  (6, 9500000000, 6935019521),
  (7, 9465000000, 8029732865),
  (8, 9465000000, 9069920257),
  (9, 10000000000, 9999999999),
  (10, 109999999999999999999999999999999999999999999999999999999999999999999999999999999999999999810, 110363899000137033701953258110096424585284286687247951331239666673017274235191757537856716801),
  (11, 119999999999999999999999999999999999999999999999999999999999999999999999999999999999999999811, 119999999999999999999999999999999999999999999999999999999999999999999999999999999999999999911),
  (12, 1296624070230872986615199999999999999999999999999999999999999999999999999999999999999999999812, 1305648619097413065366655110365318939881180944043074758425639551782425962758127732727077666817),
  (13, 13912132578246943783965088149390179999999999999999999999999999999999999999999999999999999999998, 14253176599427927419277198446645075590255589477254440739573558667130924921044223482548572913665),
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  (777, 946500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 778196962476846497303419250281035877407606694679792840112649846883012072435678855126285556114836687573708078593716502430824167540303271355117143076969272171130763658471606922208168882848992566156696434302822978877152949347741171615244802481763695844969376036851950319222478766726822053874230113753450601510231701009522412389715469094520450464051738106116026712996699517863605420257681986024232695971250617426302024809698540719947325479758989909095205934112191394146363038411587011695308909564386038981000751671949779449712713606540068414731670516027847762783109995366837232522468643171193367872019155906767426943558724320629031840211806822838491361006129621602303777840152373655879873119962889473720361572579933445400389645976454642508869607384217772347262324196807491692283362840322714978512463123632160107013392837008035764705326406146598366579092130647783311843434864597184231599667822600852094491923712686190672691716253952098383874442649205338408690314653333507603501189312648106930737694229135361),
  (800, 800802213512135806924597141583319937935661524586995796416956543180535950593308449565437207253087190466749218139212855436557526293237983838007531233982713908996746768681561476522754148459283501106260566392656302458388570988194710265427055559056886670890852349799540299711901679652382549824988303282689771084309515397157402144248386359532434178359722506367764784402688256008593729698332531289696085922661289285439999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998, 800879350819490062255090530586254213428750727732420796864996991970927248879877898353840684552987298078899210790109521431719693573261652889220708978417170282267790201435510218443400524133952502099332226507875173129431566759854796460656110935967066787992249852893668580870621626474924664357734857714902822205953523546863501379603143504308723311509343929183572853678998240005529683895274463036135934079423845612229986782476301025422820644734276992547500313114865401298199073706035632549641622588630853492328527604426574799248955743772417148312101951234327597483140972473645089357350862038605087120584394522488054291276889388819962431527609725499954915571160554973451364967798028016006926539413655014172353530791221537679559916201493297910585755743062078665010504812225781189786980468858882096080277607937401950866738526423549220760220309736410811714513603661846785104416845766072593937031204309054306480272932942921272307811117533097384191357772368188978153023087519044054315113117580517380426405374853121),
  (888, 946478274400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 889155161366616097974866005728296561725938707996217260768478762757511499424211974051647120104567436526800360556229884926614592306843910105501532480379613504084225886468301622190212497468267090574002000209911092475159512527026447524077048596719440461937360670756473509571728186291960844402172455685597899390848873849685764886101024852539861436695494284342321032993646818434739383076255279137780111120637140347961621323954885576393921763962290509473262715795830557642122516858352183683673514104583410546975331886031966370538679460645043759279831116305319916252828901542550193167467740902483318825405941966562307283371015965625293800870146021515528969317431223345069956669912155076171250149771747919824863741383229960735595467336449014806541778242551346672040268695146635456294077500804015371070625817795333897557048421544424008880257182935751833493863891612037579299824457180513679334526496425349565605491278328924637340161470134951709208858113350801795411215449566553708628339845984496461143402931552257),
  (900, 900902616337419472638171855447588335446577171445546850264624234333373072123005423482664557088114812864521766300788397747227572002296556011590053852860724732413465800218116889545782153962591281261241154391542643999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998, 900980078832721036008163965484774564426565079722370502397951105542007148678783381702197701330166068182961354590000330165593909422429076304466812289821167031848935968486439074875686935122288101455788093234370619796468799442294413311858100324356667285813177852916679882532659327107768933539216528705249179203344636938846303163266740041975696867952701491889355042179994304495345810453614322656985055084475047745834054514764771639599509805572974068194409530684464966773554201668014187375348749472203835614401877845915322450296404672260547824436150880621250766306196237642926706725343731116426751821777368535348382306985746661833515297777351292773762356269561557161289533063547314026687195214933509859679425252139172908913131995264854537395050142731895937472407328059575938339878751920332145813004313102540165721053428147538431988239816625561868029466333339528699869458790751205476953562073554578402034123980135181580992688713891744841497964885523884033259931100883020636759013660469614041681748872984002561),
  (999, 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999),
  ]

# Compute this for more digit strings:
if not "SKIP":
  for digits in (114, 115, 116, 117):
    mues, mugs = multidigit_unbounded_eq_search(digits, sizebound=1100)
    print(f"({digits}, {mues}, {mugs}),")

# Verify the above for small values:
for digits, paper_eq, paper_gt in multidigit_table[:15]:
  mues, mugs = multidigit_unbounded_eq_search(digits)
  assert paper_eq == mues, f"{paper_eq=} {mues=}"
  assert paper_gt == mugs, f"{paper_gt=} {mugs=}"
  print(f"{digits}\t{mues}\t{mugs}")

"""### Table `multidigit_num_exactly_equals`"""

if  "SKIP":
  for digits in range(1, 2):
    some_eq, n_found, reached = multidigit_all_eq_search(digits, sizebound=(223 if digits < 100 else 2023), first_and_last=True)
    print(f"({digits}, {n_found}, {some_eq}),")

multidigit_num_exactly_equals = [
(1, 82, [199981, 1111111110]),
(2, 13, [28263827, 10535000000]),
(3, 35, [371599983, 20500000000]),
(4, 47, [499999984, 30500000000]),
(5, 4, [10000000000, 40000000000]),
(6, 71, [9500000000, 59628399995]),
(7, 48, [9465000000, 69971736170]),
(8, 343, [9465000000, 79998399997]),
(9, 8, [10000000000, 80000000000]),
(10, 3167, [109999999999999999999999999999999999999999999999999999999999999999999999999999999999999999810, 5352172560000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(11, 9043, [119999999999999999999999999999999999999999999999999999999999999999999999999999999999999999811, 110103052604374646377746105259712939374508292853576387433161699999999999999999999999999999999999999910]),
(12, 7485, [1296624070230872986615199999999999999999999999999999999999999999999999999999999999999999999812, 112040626845499999999999999999999999999999999999999999999999999999999999999999999999999999999999999911]),
(13, 1305, [13912132578246943783965088149390179999999999999999999999999999999999999999999999999999999999998, 105352172560000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(14, 5299, [14999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999814, 114082202065499999999999999999999999999999999999999999999999999999999999999999999999999999999999999913]),
(15, 297, [159131124070230872986615670413207341252818821798636523899999999999999999999999999999999999999998, 115103052604374646377746105259714402920638292853576387433161699999999999999999999999999999999999999914]),
(16, 4659, [169889999999999999999999999999999999999999999999999999999999999999999999999999999999999999999816, 116122839779999999999999999999999999999999999999999999999999999999999999999999999999999999999999999915]),
(17, 1019, [1787065499999999999999999999999999999999999999999999999999999999999999999999999999999999999999817, 105352000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(18, 37, [1893472135576145619210670841910609508292853576387433181699999999999999999999999999999999999999998, 105352000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(19, 2019, [1999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999819, 119185361296624070230872986616614402920638292853576387433185599999999999999999999999999999999999999918]),
(20, 617, [20839779999999999999999999999999999999999999999999999999999999999999999999999999999999999999999820, 120203025361058275516552766837776305390186221025289395917916171225701511546423327243163416035737711661]),
(21, 621, [26000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 210103052604374646377746105259714402921478113973986731088957728043327801546423327243163416035737711661]),
(22, 1518, [22839779999999999999999999999999999999999999999999999999999999999999999999999999999999999999999822, 220206212823249411925872986616614402922099366857776133739300707174962343022875423045388830661729122414]),
(23, 629, [23893472239032076776380572105892182776227367513201355831699999999999999999999999999999999999999998, 223164521516624070235888193104034880122099366857776133739300707174962367815839729406402755637787362521]),
(24, 917, [26000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 205350000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(25, 926, [25999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999825, 225206212824374646377746105259882258261222313984229879430613195797444181745350896360550699523893656630]),
(26, 23, [267680647749411926497175976723628298763708909736395917916171233036816898408048747430432972495239800, 226226510361058275516552766837776305390186227367513201355831699999999999999999999999999999999999999998]),
(27, 1217, [278106406625234450620513245771028764999999999999999999999999999999999999999999999999999999999999827, 227246839779999999999999999999999999999999999999999999999999999999999999999999999999999999999999999926]),
(28, 35, [288521516625234450620513245771028908577480896729798636524607174962419999999999999999999999999999998, 228267680647749411927126881341685431842838113973986731088957728442805389695092289071533476933970442909]),
(29, 26, [326000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 229288521516625234450620513245771029512285594870782798636524607174962419999999999999999999999999999998]),
(30, 623, [309361303648653769055590861052676481602512754506597280692271233063922184115603588986028295193171431, 205350000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(31, 835, [319787065499999999999999999999999999999999999999999999999999999999999999999999999999999999999999831, 305350000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(32, 816, [332319339999999999999999999999999999999999999999999999999999999999999999999999999999999999999999832, 320206212933281827159111806837776311450925594870782798636524607174962419999999999999999999999999999998]),
(33, 415, [350000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 305350000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(34, 819, [349999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999834, 334288521516625234752497175976723628908577480896730462855832407174962419999999999999999999999999999998]),
(35, 15, [3566627934374646377746105273564460847223708909736395917916171233775908444839508251706929454074470879, 335309361307749411927126881352340832542347480896730462855832407174962419999999999999999999999999999998]),
(36, 411, [3669787065499999999999999999999999999999999999999999999999999999999999999999999999999999999999999836, 336330212934374646377746105273623655245992317890158558056720249788978532618493319908708364083798033012]),
(37, 411, [3772946197749411927126881355485433377900626729217201362776028766148862328336261999999999999999999998, 337349999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999936]),
(38, 807, [3875999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999838, 338370839779999999999999999999999999999999999999999999999999999999999999999999999999999999999999999937]),
(39, 7, [5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 305000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(40, 411, [4082319351125235657999999999999999999999999999999999999999999999999999999999999999999999999999999840, 340406103125765751093391017127311843999999999999999999999999999999999999999999999999999999999999999998]),
(41, 509, [4185372065499999999999999999999999999999999999999999999999999999999999999999999999999999999999999841, 405000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(42, 514, [4288521516625235657999999999999999999999999999999999999999999999999999999999999999999999999999999842, 420206212934386164263552824023276311450925594870782798637223971233850381546426770137143272180864534451]),
(43, 118, [4391680647749423853126881355485437335552512754506597280692271233850381546431256543408249035583016487, 430309372065499999999999999999999999999999999999999999999999999999999999999999999999999999999999999942]),
(44, 114, [4484412446538322897143536351473992488586139686206361296720249788978532618493319908708366019135416730, 440412521516625235657999999999999999999999999999999999999999999999999999999999999999999999999999999943]),
(45, 514, [4587467714488482170379473383385446922261689457776134486181699999999999999999999999999999999999999998, 445412521516625235657999999999999999999999999999999999999999999999999999999999999999999999999999999944]),
(46, 514, [4690627934488482170379473383385446969323145832480134486181699999999999999999999999999999999999999998, 446433372065499999999999999999999999999999999999999999999999999999999999999999999999999999999999999945]),
(47, 514, [4793787065499999999999999999999999999999999999999999999999999999999999999999999999999999999999999847, 447453159132249423853126881355485447573024968142223865506873971233850381547124571954147929806462427713]),
(48, 514, [4896840867749423853126881355488155396298114038953664219307728766148922184159819908708366019135416730, 448473999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999947]),
(49, 509, [4999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999849, 405000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(50, 4, [100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(51, 516, [51525999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999851, 510103159132250576146873118644511844603701885961046335780692271233851077815840180091291633980864583267]),
(52, 610, [95000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 520206212934499999999999999999999999999999999999999999999999999999999999999999999999999999999999999951]),
(53, 516, [53566627934499999999999999999999999999999999999999999999999999999999999999999999999999999999999999853, 530309372065511517829620526616614553030676854167519865513818299999999999999999999999999999999999999998]),
(54, 516, [54587478483374764341999999999999999999999999999999999999999999999999999999999999999999999999999999854, 540412532285511517829620526616614553077738310542223865513818299999999999999999999999999999999999999998]),
(55, 16, [95000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 550515587553461677102856463648526007511413860313793638703279750211021467381506680091291633980864583267]),
(56, 10, [95000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(57, 516, [57557129010532097269056158849999999999999999999999999999999999999999999999999999999999999999999999998, 557557129010532097269056158849999999999999999999999999999999999999999999999999999999999999999999999998]),
(58, 510, [95000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(59, 510, [95000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(60, 17, [60609180314234248906609351641181419549429228788182481861638008362722329127302243456865607042078098673, 595000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(61, 1312, [61629160219999999999999999999999999999999999999999999999999999999999999999999999999999999999999999861, 610103160219999999999999999999999999999999999999999999999999999999999999999999999999999999999999999960]),
(62, 618, [62649999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999862, 600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(63, 612, [95000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(64, 25, [64690638692250588072873118647659167457652519103269537144167592825037579999999999999999999999999999998, 640412532285625353622253894726435539152776291090263604082083828766224091555160491748293070545925529117]),
(65, 19, [65711478483374765247502824023276371091422519103269537144167592825037579999999999999999999999999999998, 650515680654760967378999999999999999999999999999999999999999999999999999999999999999999999999999999998]),
(66, 624, [94650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(67, 1224, [94650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(68, 1254, [94650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(69, 1248, [94650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(70, 62, [70711478483374765549379486754228970487714405129217201363475392825037579999999999999999999999999999998, 699674000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(71, 177, [94650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 710103160219999999999999999999999999999999999999999999999999999999999999999999999999999999999999999970]),
(72, 2842, [72753160219999999999999999999999999999999999999999999999999999999999999999999999999999999999999999872, 700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(73, 1470, [73773489638941724483447233162223694609813772632486798644168299999999999999999999999999999999999999998, 700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(74, 750, [74793787175625353622253894740117741738777686015770120569386804202555818254649103639449300476106343367, 740412532285625353622253894740117741738777686015770120569386804202555818254649103639449300476106343367]),
(75, 2143, [94650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 750515691417755648906670789716899999999999999999999999999999999999999999999999999999999999999999999998]),
(76, 1570, [76835478483375929764111806895965119877900633142223866260699292825037632184160270593597244362212637476, 760618745219999999999999999999999999999999999999999999999999999999999999999999999999999999999999999975]),
(77, 3542, [94650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(78, 1449, [94650000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 700000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(79, 2164, [79796974638941724483447233162223694609813778974710604082083828774298488453576672756836583964262288335, 779796974638941724483447233162223694609813778974710604082083828774298488453576672756836583964262288335]),
(80, 4847, [80814638703375929769127013383385597079361707146423612566814399999999999999999999999999999999999999880, 799980212934499999999999999999999999999999999999999999999999999999999999999999999999999999999999999979]),
(81, 252, [94648000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 810103160219999999999999999999999999999999999999999999999999999999999999999999999999999999999999999980]),
(82, 4080, [94648000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 800000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(83, 18740, [83877160219999999999999999999999999999999999999999999999999999999999999999999999999999999999999999883, 830309373154499999999999999999999999999999999999999999999999999999999999999999999999999999999999999982]),
(84, 1292, [84896947395625353622253894740285597079361707146423612566838299999999999999999999999999999999999999884, 840412532285625353622253894740285597079361707146423612566838299999999999999999999999999999999999999983]),
(85, 13121, [85917797934499999999999999999999999999999999999999999999999999999999999999999999999999999999999999885, 850515691417864423857502829716899999999999999999999999999999999999999999999999999999999999999999999998]),
(86, 5324, [94647827440000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 860618745219999999999999999999999999999999999999999999999999999999999999999999999999999999999999999985]),
(87, 30044, [87959373154499999999999999999999999999999999999999999999999999999999999999999999999999999999999999887, 870721893593375929769127013384799999999999999999999999999999999999999999999999999999999999999999999986]),
(88, 19523, [94647827440000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 880825053813375929769127013384799999999999999999999999999999999999999999999999999999999999999999999987]),
(89, 22873, [89898515576351346230944409138966940227406178635456602431024816768256684368764757045898760580908855464, 889898515576351346230944409138966940227406178635456602431024816768256684368764757045898760580908855464]),
(90, 29879, [90917797934499999999999999999999999999999999999999999999999999999999999999999999999999999999999999909, 899999999901125353622253903451171012061360633142223872993126099999999999999999999999999999999999999998]),
(91, 32148, [94647827439076400000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 910103160219999999999999999999999999999999999999999999999999999999999999999999999999999999999999999990]),
(92, 108109, [92959373154499999999999999999999999999999999999999999999999999999999999999999999999999999999999999929, 920206319352250589291726443451171012061360633142223872993126099999999999999999999999999999999999999998]),
(93, 133605, [93980212937409627214391585751759227762596291735756999999999999999999999999999999999999999999999999998, 900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000]),
(94, 209755, [94999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999949, 940412532286750589401111812590117802427037550961703999999999999999999999999999999999999999999999999998]),
    ]

print([n for d, n, some in multidigit_num_exactly_equals])

[(d, n, n % d) for d, n, some in multidigit_num_exactly_equals]

"""### Largest E_="""

# https://oeis.org/draft/A364970
[values[-1] for d, count, values in multidigit_num_exactly_equals]

"""### Where a= is larger than a>, or equivalently a≥ = a>"""

# The sequence of multi-digit strings where a_=(d) > a_\ge(d) in base 10.
# https://oeis.org/draft/A364969
[digits for digits, eq, gt in multidigit_table if eq > gt]

"""### Number of digits in a="""

for d, eq, gt in multidigit_table:
  print(f"({d}, {len(str(eq))}),")

"""### Starting digits in a="""

first_digits = [int(str(eq)[:len(str(d))]) for d, eq, gt in multidigit_table]
print(first_digits)

"""### Sorted by a=, by a>, and by their difference"""

from collections import defaultdict
by_eq = defaultdict(list)
by_gt = defaultdict(list)
by_df = defaultdict(list)
by_dp = defaultdict(list)
for digits, eq, gt in multidigit_table:
  by_eq[eq].append(digits)
  by_gt[gt].append(digits)
  by_df[gt-eq].append(digits)
  overload = len(str(gt))-4
  trunc_eq, trunc_gt = eq // 10**overload, gt // 10**overload
  by_dp[round(100*((trunc_gt-trunc_eq)/(1.0*trunc_gt)), 0)].append(digits)
def print_by_numkey(name, dct):
  for key, values in sorted(dct.items()):
    print(f"{name}| {key}\t{values}")
def print_by_length(name, dct):
  for key, values in sorted(dct.items(), key=lambda i: len(i[1])):
    print(f"{name}| {len(values)}\t{key}\t{values}")
for name, dct in [("by_eq", by_eq),
                  ("by_gt", by_gt),
                  ("by_df", by_df),
                  ("by_d%", by_dp)]:
  print_by_numkey(name, dct)
  print_by_length(name, dct)

"""## All Your Base Are Belong to Us!

Note that the unbounded search with safeleft=2 above only works for bases above 2, because for base 2, 2 is not in fact safeleft.

Base 1:
* There are no zeroes
* 1 uses one 1, but we're not counting that.
* 2 is a_>, and it's clear there can never be an a_= above 1.

Base 2, Counting Zeroes:
* 1: uses 0, saving 1
* 10: uses 1, 1 saved.
* 11: uses 0, 2 saved now.
* 100: uses 2, but got one with this tape, so 1 saved now.
* 101: uses 1, 1 saved.
* 110: uses 1, 1 saved.
* 111: uses 0, 2 saved.
* 1000: uses 3, got one with this tape, so 8 is a_=.
* 1001: uses 2, got only one, so 9 is a_>.

Base 2, Counting Ones:
* 1 is = but doesn't count.
* 10 uses 1, got one, so this is a_=.
* 11 uses 2, got only one, and we were just out, so this is a_>.

Base 3, counting 0:
* 1 saves the first zero.
* 2 saves the second zero.
* 10 uses a zero but also gets one, so still two saved.
* Clearly 2 is safeleft.

Base 3, couting 1:
* 1 is = but doesn't count.
* 2 saves 1x1.
* 10 uses 1x1, still 1x1 saved.
* Clearly 2 is safeleft.

Base 3, counting 2:
* 1 saves 1x2
* 2 still have 1x2 saved.
* 10 saves another, so we have 2x2 saved.
* Clearly 2 is safeleft.

And so, for base>2, 2 is safeleft.
"""

def letterbase(value, base):
  tr = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyzΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩαβγδεζηθικλμνξοπρςστυφχψω"
  if base > len(tr):
    raise ValueError(f"base {base} is out of range.")
  return ''.join([tr[i] for i in strbase(value, base)])+"_{"+str(base)+"}"
assert "A392EFAB_{16}" == letterbase(0xA392EFAB, 16)
assert "16_{10}" == letterbase(0x10, 10)

"""### `table_bases` for bases 1-49:"""

table_bases = [ (1, 1, None, 2),
                (2, 0, 8, 9),
                (2, 1, 2, 3),
                (3, 0, None, 87),
                (3, 1, 4, 13),
                (3, 2, 27, 26),
                (4, 0, None, 1068),
                (4, 1, 25, 29),
                (4, 2, 256, 170),
                (4, 3, 256, 255),
                (5, 0, None, 16022),
                (5, 1, 181, 182),
                (5, 2, 3125, 1562),
                (5, 3, 2787, 2343),
                (5, 4, 3125, 3124),
                (6, 0, None, 284704),
                (6, 1, 421, 427),
                (6, 2, 3878, 3884),
                (6, 3, 46656, 27993),
                (6, 4, 42768, 37324),
                (6, 5, 46656, 46655),
                (7, 0, None, 5834024),
                (7, 1, 3930, 3931),
                (7, 2, 46340, 46341),
                (7, 3, 823543, 411771),
                (7, 4, 761729, 549028),
                (7, 5, 761664, 686285),
                (7, 6, 823543, 823542),
                (8, 0, None, 135430302),
                (8, 1, 8177, 8185),
                (8, 2, 679922, 679930),
                (8, 3, 1048563, 1048571),
                (8, 4, 16777216, 9586980),
                (8, 5, 15728640, 11983725),
                (8, 6, 15638528, 14380470),
                (8, 7, 16777216, 16777215),
                (9, 0, None, 3511116537),
                (9, 1, 102772, 102781),
                (9, 2, 1594307, 1594316),
                (9, 3, 22320522, 18295908),
                (9, 4, 387420489, 193710244),
                (9, 5, 365099967, 242137805),
                (9, 6, 365099967, 290565366),
                (9, 7, 364957812, 338992927),
                (9, 8, 387420489, 387420488),
                (10, 0, None, 100559404366),
                (10, 1, 199981, 199991),
                (10, 2, 28263827, 28263828),
                (10, 3, 371599983, 371599993),
                (10, 4, 499999984, 499999994),
                (10, 5, 10000000000, 5555555555),
                (10, 6, 9500000000, 6666666666),
                (10, 7, 9465000000, 7777777777),
                (10, 8, 9465000000, 8888888888),
                (10, 9, 10000000000, 9999999999),
                (11, 0, 3152738985031, 3152738985032),
                (11, 1, 3179142, 3179143),
                (11, 2, 581404546, 581404547),
                (11, 3, 13344870724, 8542241260),
                (11, 4, 11484304153, 11484304154),
                (11, 5, 285311670611, 142655835305),
                (11, 6, 271912181709, 171187002366),
                (11, 7, 271912157896, 199718169427),
                (11, 8, 271912157751, 228249336488),
                (11, 9, 271912157751, 256780503549),
                (11, 10, 285311670611, 285311670610),
                (12, 0, None, 107400330425888),
                (12, 1, 5971945, 5971957),
                (12, 2, 1289945066, 1289945078),
                (12, 3, 20639121387, 20639121399),
                (12, 4, 295916986366, 295916986367),
                (12, 5, 371504185325, 371504185337),
                (12, 6, 8916100448256, 4863327517230),
                (12, 7, 8544596262912, 5673882103435),
                (12, 8, 8544596262912, 6484436689640),
                (12, 9, 8523957141504, 7294991275845),
                (12, 10, 8522577616896, 8105545862050),
                (12, 11, 8916100448256, 8916100448255),
                (13, 0, 3950024143546664, 3950024143546665),
                (13, 1, 143610511, 114818731),
                (13, 2, 30411771424, 30411771425),
                (13, 3, 535464000311, 535464000312),
                (13, 4, 11959840812819, 8424009892964),
                (13, 5, 10568722104311, 10568722104312),
                (13, 6, 302875106592253, 151437553296126),
                (13, 7, 290915265779434, 176677145512147),
                (13, 8, 290915265779434, 201916737728168),
                (13, 9, 290915265779419, 227156329944189),
                (13, 10, 290915265776631, 252395922160210),
                (13, 11, 290915265776631, 277635514376231),
                (13, 12, 302875106592253, 302875106592252),
                (14, 0, None, 155996847068247395),
                (14, 1, 210826981, 210826995),
                (14, 2, 803108615022, 803108615023),
                (14, 3, 15393391704939, 15393391704940),
                (14, 4, 262803244417256, 262803244417270),
                (14, 5, 329699565906945, 329699565906946),
                (14, 6, 396857386627050, 396857386627064),
                (14, 7, 11112006825558016, 5983388290685085),
                (14, 8, 10715149438930944, 6838158046497240),
                (14, 9, 10715149438930944, 7692927802309395),
                (14, 10, 10695883012804864, 8547697558121550),
                (14, 11, 10695883012804864, 9402467313933705),
                (14, 12, 10695883012804864, 10257237069745860),
                (14, 13, 11112006825558016, 11112006825558015),
                (15, 0, None, 6584073072068125453),
                (15, 1, 4754446846, 4754446861),
                (15, 2, 1729951171847, 1729951171862),
                (15, 3, 648731689453125, 483487477377923),
                (15, 4, 648731689453099, 648731689453114),
                (15, 5, 14920828857421875, 11193615385790624),
                (15, 6, 13458130908046873, 13458130908046874),
                (15, 7, 437893890380859375, 218946945190429687),
                (15, 8, 422973061523437500, 250225080217633928),
                (15, 9, 422973061523437500, 281503215244838169),
                (15, 10, 422973061523437500, 312781350272042410),
                (15, 11, 422973061523437500, 344059485299246651),
                (15, 12, 422940192451171875, 375337620326450892),
                (15, 13, 422940093447238521, 406615755353655133),
                (15, 14, 437893890380859375, 437893890380859374),
                (16, 0, 295764262988176583799, 295764262988176583800),
                (16, 1, 8589934561, 8589934577),
                (16, 2, 50938312130530, 50938312130546),
                (16, 3, 1125899906842595, 1125899906842611),
                (16, 4, 22134509096927204, 22134509096927220),
                (16, 5, 576460752303423488, 410202866059662677),
                (16, 6, 493144159197069286, 493144159197069302),
                (16, 7, 576460752303423463, 576460752303423479),
                (16, 8, 18446744073709551616, 9838263505978427528),
                (16, 9, 17870283321406128128, 11068046444225730969),
                (16, 10, 17870283321406128128, 12297829382473034410),
                (16, 11, 17845513523455590400, 13527612320720337851),
                (16, 12, 17845513523455590400, 14757395258967641292),
                (16, 13, 17845454776892915712, 15987178197214944733),
                (16, 14, 17845454776892915712, 17216961135462248174),
                (16, 15, 18446744073709551616, 18446744073709551615),
                (17, 0, None, 14088968131538370019982),
                (17, 1, 222195898593, 222195898594),
                (17, 2, 1641463484416521, 1641463484416522),
                (17, 3, 38769113643155798, 38769113643155799),
                (17, 4, 968710025627986624, 815458547947371445),
                (17, 5, 24793200517348288362, 16151057595869037000),
                (17, 6, 23868089390764080488, 19425270196610623214),
                (17, 7, 22692220954727290250, 22692220954727290251),
                (17, 8, 827240261886336764177, 413620130943168382088),
                (17, 9, 802403560503060042067, 465322647311064429849),
                (17, 10, 802403560503060042067, 517025163678960477610),
                (17, 11, 802403560502351968712, 568727680046856525371),
                (17, 12, 802403462733605477968, 620430196414752573132),
                (17, 13, 802403462469942734159, 672132712782648620893),
                (17, 14, 802403462469942728252, 723835229150544668654),
                (17, 15, 802403462469942728252, 775537745518440716415),
                (17, 16, 827240261886336764177, 827240261886336764176),
                (18, 0, None, 709394716006812244474473),
                (18, 1, 396718580701, 396718580719),
                (18, 2, 3470494144278494, 3470494144278512),
                (18, 3, 1440932130947471614, 1440932130947471615),
                (18, 4, 32291161192024068064, 32291161192024068082),
                (18, 5, 40479843698864750561, 40479843698864750579),
                (18, 6, 817785453337304702946, 817785453337304702964),
                (18, 7, 955138986905903640547, 955138986905903640565),
                (18, 8, 1092955779869348265956, 1092955779869348265974),
                (18, 9, 39346408075296537575424, 20830451333980519892871),
                (18, 10, 38253452295427189309440, 23144945926645022103190),
                (18, 11, 38253452295427189309440, 25459440519309524313509),
                (18, 12, 38253452295427189309440, 27773935111974026523828),
                (18, 13, 38212972451728324558848, 30088429704638528734147),
                (18, 14, 38211247616138618130432, 32402924297303030944466),
                (18, 15, 38211247616138618130432, 34717418889967533154785),
                (18, 16, 38211247447136502736896, 37031913482632035365104),
                (18, 17, 39346408075296537575424, 39346408075296537575423),
                (19, 0, None, 37644984315968494382106708),
                (19, 1, 13494919482970, 11575488191148),
                (19, 2, 122536147345585331, 122536147345585332),
                (19, 3, 57465296623637492013, 57465296623637492014),
                (19, 4, 1365193971372874995805, 1365193971372874995806),
                (19, 5, 1711104305065560619656, 1711104305065560619657),
                (19, 6, 36651543627512253609944, 36651543627512253609945),
                (19, 7, 42808006784347146759384, 42808006784347146759385),
                (19, 8, 51126972052076023422525, 48977572837968997034693),
                (19, 9, 1978419655660313589123979, 989209827830156794561989),
                (19, 10, 1925419277414476925965579, 1099122030922396438402210),
                (19, 11, 1925419277414476925965579, 1209034234014636082242431),
                (19, 12, 1925416213358319388167265, 1318946437106875726082652),
                (19, 13, 1925416213358319388167265, 1428858640199115369922873),
                (19, 14, 1925416213358319320815916, 1538770843291355013763094),
                (19, 15, 1925416213358319320815916, 1648683046383594657603315),
                (19, 16, 1925416213358289614730210, 1758595249475834301443536),
                (19, 17, 1925416213358289614730210, 1868507452568073945283757),
                (19, 18, 1978419655660313589123979, 1978419655660313589123978),
                (20, 0, None, 2099915447874594268014136006),
                (20, 1, 20479999999961, 20479999999981),
                (20, 2, 4662353919999999962, 4662353919999999982),
                (20, 3, 131071999999999999963, 131071999999999999983),
                (20, 4, 3276799999999999999964, 3276799999999999999984),
                (20, 5, 88473600000000000000000, 76981662740479999999999),
                (20, 6, 1742426678565430991331998, 1742426678565430991331999),
                (20, 7, 2035723568627669509670662, 2035723568627669509670663),
                (20, 8, 2328145353256959999999968, 2328145353256959999999988),
                (20, 9, 2621439999999999999999969, 2621439999999999999999989),
                (20, 10, 104857600000000000000000000, 55188210526315789473684210),
                (20, 11, 102236160000000000000000000, 60707031578947368421052631),
                (20, 12, 102236160000000000000000000, 66225852631578947368421052),
                (20, 13, 102236160000000000000000000, 71744673684210526315789473),
                (20, 14, 102147686400000000000000000, 77263494736842105263157894),
                (20, 15, 102147686400000000000000000, 82782315789473684210526315),
                (20, 16, 102147686400000000000000000, 88301136842105263157894736),
                (20, 17, 102147549705994240000000000, 93819957894736842105263157),
                (20, 18, 102147549705994240000000000, 99338778947368421052631578),
                (20, 19, 104857600000000000000000000, 104857599999999999999999999),
                (21, 0, None, 122840510450378925218598549446),
                (21, 1, 764527028941797, 665306762187614),
                (21, 2, 9731759797564526003, 9731759797564526024),
                (21, 3, 5624957162992296052815, 5624957162992296052836),
                (21, 4, 148302287555085812369239, 148302287555085812369260),
                (21, 5, 3673223540311337661352219, 3673223540311337661352220),
                (21, 6, 4416165546777008708526891, 4416165546777008708526912),
                (21, 7, 102313412417944368945533122, 102313412417944368945533123),
                (21, 8, 117009630932338884853362556, 117009630932338884853362557),
                (21, 9, 136901131950087269964334737, 131743265100242585924033124),
                (21, 10, 5842587018385982521381124421, 2921293509192991260690562210),
                (21, 11, 5701269720889118242708262757, 3213422860112290386759618431),
                (21, 12, 5701269720889118242708262757, 3505552211031589512828674652),
                (21, 13, 5701269720889118242708262757, 3797681561950888638897730873),
                (21, 14, 5701269720889118242708262757, 4089810912870187764966787094),
                (21, 15, 5701269720889118242708262757, 4381940263789486891035843315),
                (21, 16, 5701107346476895878591235302, 4674069614708786017104899536),
                (21, 17, 5701107346476895878591235302, 4966198965628085143173955757),
                (21, 18, 5701107346476895878591235302, 5258328316547384269243011978),
                (21, 19, 5701107346064205010924641609, 5550457667466683395312068199),
                (21, 20, 5842587018385982521381124421, 5842587018385982521381124420),
                (22, 0, None, 7519552201187745418233309682258),
                (22, 1, 1168636602822613, 1168636602822635),
                (22, 2, 400576288567329819579, 400576288567329819580),
                (22, 3, 256562055568599073615831, 256562055568599073615853),
                (22, 4, 7113485943120469933044690, 7113485943120469933044691),
                (22, 5, 185272001789229063619903894, 185272001789229063619903895),
                (22, 6, 222653645244124605930631646, 222653645244124605930631647),
                (22, 7, 5420782343751988814428827611, 5420782343751988814428827633),
                (22, 8, 6200573675557158331361790333, 6200573675557158331361790334),
                (22, 9, 6979449004912524345039695870, 6979449004912524345039695871),
                (22, 10, 7759724485550444486287425502, 7759724485550444486287425524),
                (22, 11, 341427877364219557396646723584, 178843173857448339588719712353),
                (22, 12, 333668152878669112910359298048, 195101644208125461369512413476),
                (22, 13, 333668152878669112910359298048, 211360114558802583150305114599),
                (22, 14, 333668152878669112910359298048, 227618584909479704931097815722),
                (22, 15, 333427665549571475168015679488, 243877055260156826711890516845),
                (22, 16, 333427665549571475168015679488, 260135525610833948492683217968),
                (22, 17, 333427665549571475168015679488, 276393995961511070273475919091),
                (22, 18, 333427367424783337600566493184, 292652466312188192054268620214),
                (22, 19, 333427366979554996020588187648, 308910936662865313835061321337),
                (22, 20, 333427366979554996020588187648, 325169407013542435615854022460),
                (22, 21, 341427877364219557396646723584, 341427877364219557396646723583),
                (23, 0, None, 480725637865418016826677007715690),
                (23, 1, 41826814261329722, 41826814261329723),
                (23, 2, 17578309949348561322559, 17578309949348561322560),
                (23, 3, 12403394925800154563160419, 12403394925800154563160420),
                (23, 4, 360668496634743777682404399, 360668496634743777682404400),
                (23, 5, 9845043097046282331104781043, 9845043097046282331104781044),
                (23, 6, 11830651372936896727808467704, 11830651372936896727808467705),
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                (48, 3, 13388888869699032553439690152857412882673530264138319180857251, 13388888869699032553439690152857412882673530264138319180857299),
                (48, 4, 1821491386278204883732702646845486735623320424784697632959737364388, 1821491386278204883732702646845486735623320424784697632959737364436),
                (48, 5, 106610687847198389386800159363706104397485436245954119629155027910565, 106610687847198389386800159363706104397485436245954119629155027910613),
                (48, 6, 280577911829581306719506055934688464957030428779000979567206388713127934, 280577911829581306719506055934688464957030428779000979567206388713127935),
                (48, 7, 327508033066593452196250089565305152709075260147571055500764245741535143, 327508033066593452196250089565305152709075260147571055500764245741535191),
                (48, 8, 17544643432509396375742870768922045874648246198309560975981227774102732712, 17544643432509396375742870768922045874648246198309560975981227774102732760),
                (48, 9, 925677489295718475061136519922734779500152295152593554145440664258634317822, 925677489295718475061136519922734779500152295152593554145440664258634317823),
                (48, 10, 1028749661976458780425295748036886358526650953038579463134874358373641355262, 1028749661976458780425295748036886358526650953038579463134874358373641355263),
                (48, 11, 1131867762278146970790240309537694607762564099070005567810641233282745761707, 1131867762278146970790240309537694607762564099070005567810641233282745761755),
                (48, 12, 57931445146664362735143039152281832913717383754888501786726830323102648369150, 57931445146664362735143039152281832913717383754888501786726830323102648369151),
                (48, 13, 62766025384671870675546874239823557522800402055005575195582187041101138886654, 62766025384671870675546874239823557522800402055005575195582187041101138886655),
                (48, 14, 67602757363120867378498304297145294103158092463134171608130573659843206315950, 67602757363120867378498304297145294103158092463134171608130573659843206315998),
                (48, 15, 72439536785801406130575379810412454896804102340480356339881038930095728754607, 72439536785801406130575379810412454896804102340480356339881038930095728754655),
                (48, 16, 3626813571782168623664415397999907196554110132895835468302589357766972331261872, 3626813571782168623664415397999907196554110132895835468302589357766972331261920),
                (48, 17, 3853803540284281885417130990213117282119007584875211917137914044258515058425854, 3853803540284281885417130990213117282119007584875211917137914044258515058425855),
                (48, 18, 4080795662556464959696580612636906260018203827355822679787840182942841922650110, 4080795662556464959696580612636906260018203827355822679787840182942841922650111),
                (48, 19, 5215646648577701241401427346349696752569895368514585656471434802966892470337536, 4307682513004213991986019691887372204998487291966523831556589550787573554610175),
                (48, 20, 4534674635230002037767509175902949259201031073350789207897132162344691076956158, 4534674635230002037767509175902949259201031073350789207897132162344691076956159),
                (48, 21, 4761664557803862827747034928974212852249393137091452463553364168003321702907829, 4761664557803862827747034928974212852249393137091452463553364168003321702907877),
                (48, 22, 5215646648577701241401427346349696752569895368514585656471434802966892470337536, 4988656376684502622470129442046746944784981925941260691676379426129725598933948),
                (48, 23, 5215646648577701241401427346349696752569895368514585656471434802966892470337463, 5215646648577701241401427346349696752569895368514585656471434802966892470337511),
                (48, 24, 500702078263459319174537025249570888246709955377400223021257741084821677152403456, 255677656985596248089125289489142581232362530405480964947025229490121707482078360),
                (48, 25, 495486431614881617933135597903221191494140060008885637364786306281854784682065920, 266330892693329425092838843217856855450377635839042671819817947385543445293831625),
                (48, 26, 495486431614881617933135597903221191494140060008885637364786306281854784682065920, 276984128401062602096552396946571129668392741272604378692610665280965183105584890),
                (48, 27, 495486431614881617933135597903221191494140060008885637364786306281854784682065920, 287637364108795779100265950675285403886407846706166085565403383176386920917338155),
                (48, 28, 495486431614881617933135597903221191494140060008885637364786306281854784682065920, 298290599816528956103979504403999678104422952139727792438196101071808658729091420),
                (48, 29, 495486431614881617933135597903221191494140060008885637364786306281854784682065920, 308943835524262133107693058132713952322438057573289499310988818967230396540844685),
                (48, 30, 495486431614881617933135597903221191494140060008885637364786306281854784682065920, 319597071231995310111406611861428226540453163006851206183781536862652134352597950),
                (48, 31, 495486431614881617933135597903221191494140060008885637364786306281854784682065920, 330250306939728487115120165590142500758468268440412913056574254758073872164351215),
                (48, 32, 495486431614881617933135597903221191494140060008885637364786306281854784682065920, 340903542647461664118833719318856774976483373873974619929366972653495609976104480),
                (48, 33, 495413992078095816527005022523410779039243255906545157008446425242924688953311232, 351556778355194841122547273047571049194498479307536326802159690548917347787857745),
                (48, 34, 495413992078095816527005022523410779039243255906545157008446425242924688953311232, 362210014062928018126260826776285323412513584741098033674952408444339085599611010),
                (48, 35, 495413992078095816527005022523410779039243255906545157008446425242924688953311232, 372863249770661195129974380504999597630528690174659740547745126339760823411364275),
                (48, 36, 495413992078095816527005022523410779039243255906545157008446425242924688953311232, 383516485478394372133687934233713871848543795608221447420537844235182561223117540),
                (48, 37, 495412860210333538380034232283101241344635493342446087002878614601691406207549440, 394169721186127549137401487962428146066558901041783154293330562130604299034870805),
                (48, 38, 495412860210333538380034232283101241344635493342446087002878614601691406207549440, 404822956893860726141115041691142420284574006475344861166123280026026036846624070),
                (48, 39, 495412841214867620517614004900596046556936636216080998443757395557365153198505984, 415476192601593903144828595419856694502589111908906568038915997921447774658377335),
                (48, 40, 495412841214867620517614004900596046556936636216080998443757395557365153198505984, 426129428309327080148542149148570968720604217342468274911708715816869512470130600),
                (48, 41, 495412841214867620517614004900596046556936636216080998443757395557365153198505984, 436782664017060257152255702877285242938619322776029981784501433712291250281883865),
                (48, 42, 495412841209004032686018093484322037791932800474219299450229918977761626663419904, 447435899724793434155969256605999517156634428209591688657294151607712988093637130),
                (48, 43, 495412841209002033162576514891402442001725992442464575219873212032898519065952256, 458089135432526611159682810334713791374649533643153395530086869503134725905390395),
                (48, 44, 495412841209002033162576514891402442001725992442464575219873212032898519065952256, 468742371140259788163396364063428065592664639076715102402879587398556463717143660),
                (48, 45, 495412841209002033162317434468034061561329210243687856618447056842963840218955776, 479395606847992965167109917792142339810679744510276809275672305293978201528896925),
                (48, 46, 495412841209002033162317432539149260688858561615632077281534080922146734790934528, 490048842555726142170823471520856614028694849943838516148465023189399939340650190),
                (48, 47, 500702078263459319174537025249570888246709955377400223021257741084821677152403456, 500702078263459319174537025249570888246709955377400223021257741084821677152403455),
                (49, 0, None, 3235164209102086433524906665277556071651973699613210041963279974766619706484733432959),
                (49, 1, 3521922612922634055450953133193962819017380, 3521922612922634055450953133193962819017429),
                (49, 2, 177415405798866612448023697696090994614426956208278621144, 177415405798866612448023697696090994614426956208278621145),
                (49, 3, 1368564295015455079991949671684722943400894950243781165203189685, 1368564295015455079991949671684722943400894950243781165203189734),
                (49, 4, 222987179439097352318352634156080207246089945918491804044546740431123, 198522821562459249906063848553852734807034902483911563699835241177204),
                (49, 5, 11867749735928663937454258812712909248032905207640403636635402926259579, 11867749735928663937454258812712909248032905207640403636635402926259628),
                (49, 6, 681292175541205709486531011694243236571309860372760091522256581907552715, 681292175541205709486531011694243236571309860372760091522256581907552764),
                (49, 7, 39514946181389931150218798678266107721135971901620085308290881750638062806, 38038398933618867151899407596640415648144842480990622298012447177375969636),
                (49, 8, 2081347596278383442481352240725913087725351623438782079600493857727573825295, 2081347596278383442481352240725913087725351623438782079600493857727573825344),
                (49, 9, 112162493443699778364186573517246652523207884932888239387492145592604233721935, 112162493443699778364186573517246652523207884932888239387492145592604233721984),
                (49, 10, 139703729639953019170179589664986435361603086038457066087279367916217055944999, 124650918475797277282637776210835848490170052259784830147696259752935131087342),
                (49, 11, 137144796228620250525348179185062857765041246202896979183521772194572287601820, 137144796228620250525348179185062857765041246202896979183521772194572287601869),
                (49, 12, 9258879891735705302921116591851812077570500981691374876803483343025476468871225, 7169297742423093295380244216820244064383296978098775227825397259077556344686832),
                (49, 13, 7767586618055216861800803844708158289203390456068643849568528706517820504073776, 7767586618055216861800803844708158289203390456068643849568528706517820504073777),
                (49, 14, 8366136312293236990877648903503846589852867194594581800664257947223780738821226, 8366136312293236990877648903503846589852867194594581800664257947223780738821227),
                (49, 15, 8964680556194876263482448548282079653766722059013922332346106815029398016747965, 8964680556194876263482448548282079653766722059013922332346106815029398016747966),
                (49, 16, 678198039435533035872498267047463012315394696650186975473401847082858922398313010, 458409457958486466795384243533883334241971758170205745246933862704942488495475809),
                (49, 17, 487098893118358548328742273328735380030244148410413586375200794437340466683398310, 487098893118358548328742273328735380030244148410413586375200794437340466683398311),
                (49, 18, 515788583762838704138452227456993136134069166033432809484414143824774663286400930, 515788583762838704138452227456993136134069166033432809484414143824774663286400931),
                (49, 19, 668939199058743511959508300674409878503931916804467502216683672030715196567504591, 544467992160738491053685029898548270538151570538191329971123659416967172712393858),
                (49, 20, 573155215689404547441667941551548847650809082014470249634657013492601815746178497, 573155215689404547441667941551548847650809082014470249634657013492601815746178498),
                (49, 21, 601844650846943184490289582011550830447861277075813315001842672226007954216304801, 601844650846943184490289582011550830447861277075813315001842672226007954216304802),
                (49, 22, 630534341491382139356652632301695218687303161762310607355637965365301735507043824, 630534341491382139356652632301695218687303161762310607355637965365301735507043873),
                (49, 23, 659223771200916817010211376932013066123596004185435415853934477978421235288424975, 659223771200916817010211376932013066123596004185435415853934477978421235288424976),
                (49, 24, 66009724686219550843768321818371771650147004059278069406814190436565131829325062449, 33004862343109775421884160909185885825073502029639034703407095218282565914662531224),
                (49, 25, 65331526646784017807895823551324308637831609362627882431340788589482272906926749439, 34380064940739349397796000947068631067784897947540661149382390852377672827773470025),
                (49, 26, 65331526646784017807895823551324308637831609362627882431340788589482272906926749439, 35755267538368923373707840984951376310496293865442287595357686486472779740884408826),
                (49, 27, 65331526646784017807895823551324308637831609362627882431340788589482272906926749439, 37130470135998497349619681022834121553207689783343914041332982120567886653995347627),
                (49, 28, 65331526646784017807895823551324308637831609362627882431340788589482272906926749439, 38505672733628071325531521060716866795919085701245540487308277754662993567106286428),
                (49, 29, 65331526646784017807895823551324308637831609362627882431340788589482272906926749439, 39880875331257645301443361098599612038630481619147166933283573388758100480217225229),
                (49, 30, 65331526646784017807895823551324308637831609362627882431340788589482272906926749439, 41256077928887219277355201136482357281341877537048793379258869022853207393328164030),
                (49, 31, 65331526646784017807895823551324308637831609362627882431340788589482272906926749439, 42631280526516793253267041174365102524053273454950419825234164656948314306439102831),
                (49, 32, 65331526646784017807895823551324308637831609362627882431340788589482272906926749439, 44006483124146367229178881212247847766764669372852046271209460291043421219550041632),
                (49, 33, 65331526607269071626505892401105509959565501641491910529720703281191391156288686633, 45381685721775941205090721250130593009476065290753672717184755925138528132660980433),
                (49, 34, 65331526607269071626505892401105509959565501641491910529720703281191391156288686633, 46756888319405515181002561288013338252187461208655299163160051559233635045771919234),
                (49, 35, 65331526607269071626505892401105509959565501641491910529720703281191391156288686633, 48132090917035089156914401325896083494898857126556925609135347193328741958882858035),
                (49, 36, 65331526607269071626505892401105509959565501641491910529720703281191391156288686633, 49507293514664663132826241363778828737610253044458552055110642827423848871993796836),
                (49, 37, 65331526607269071626505892401105509959565501641491910529720703281191391156288686633, 50882496112294237108738081401661573980321648962360178501085938461518955785104735637),
                (49, 38, 65331526607269071626505892401105509959565501641491910529720703281191391156288686633, 52257698709923811084649921439544319223033044880261804947061234095614062698215674438),
                (49, 39, 65331526607269071626505892401105509959565501641491910529720703281191391156288686633, 53632901307553385060561761477427064465744440798163431393036529729709169611326613239),
                (49, 40, 65331526607269071626505892401105509959565501641491910529720703281191391156288686633, 55008103905182959036473601515309809708455836716065057839011825363804276524437552040),
                (49, 41, 65331526607269071626505892401105509959565501641491910529720703281191391156288686633, 56383306502812533012385441553192554951167232633966684284987120997899383437548490841),
                (49, 42, 65331526607269071626505892401105509959565501641491910529720703281191391156288686633, 57758509100442106988297281591075300193878628551868310730962416631994490350659429642),
                (49, 43, 65331526607256712066554122897136998731313417208703827613996474084021341059491058878, 59133711698071680964209121628958045436590024469769937176937712266089597263770368443),
                (49, 44, 65331526607256712066554122897136998731313417208703827613996474084021341059491058878, 60508914295701254940120961666840790679301420387671563622913007900184704176881307244),
                (49, 45, 65331526607256712066554122897136998731313417208703827613996474084021341059491058878, 61884116893330828916032801704723535922012816305573190068888303534279811089992246045),
                (49, 46, 65331526607256712066554113193534835197922893844830141204351685174484465843587208777, 63259319490960402891944641742606281164724212223474816514863599168374918003103184846),
                (49, 47, 65331526607256712066554113193534835197922893844830141204351685174484465843587208777, 64634522088589976867856481780489026407435608141376442960838894802470024916214123647),
                (49, 48, 66009724686219550843768321818371771650147004059278069406814190436565131829325062449, 66009724686219550843768321818371771650147004059278069406814190436565131829325062448),
]

"""### `table_bases` for bases 50-80"""

table_bases += [
                (50, 0, 444179859561011965929496863186893220413478345535397637990204496296663272376585291071790, 444179859561011965929496863186893220413478345535397637990204496296663272376585291071791),
                (50, 1, 5960464477539062499999999999999999999999901, 5960464477539062499999999999999999999999951),
                (50, 2, 17109372010454535484313964843749999999999999999999999999902, 17109372010454535484313964843749999999999999999999999999952),
                (50, 3, 143538422485993532463908195495605468749999999999999999999999999903, 143538422485993532463908195495605468749999999999999999999999999953),
                (50, 4, 454747350886464118957519531249999999999999999999999999999999999999904, 454747350886464118957519531249999999999999999999999999999999999999954),
                (50, 5, 1353099545785943073174562542257678823062231945847507007974710114201295635, 1353099545785943073174562542257678823062231945847507007974710114201295636),
                (50, 6, 79308959404342124682414955452656745910644531249999999999999999999999999906, 79308959404342124682414955452656745910644531249999999999999999999999999956),
                (50, 7, 4520942865096286233605500310659408569335937499999999999999999999999999999907, 4520942865096286233605500310659408569335937499999999999999999999999999999957),
                (50, 8, 252556663098115960430444832891225814819335937499999999999999999999999999999908, 252556663098115960430444832891225814819335937499999999999999999999999999999958),
                (50, 9, 284217094304040074348449707031249999999999999999999999999999999999999999999909, 284217094304040074348449707031249999999999999999999999999999999999999999999959),
                (50, 10, 15442030112010845085854154575294808125495910644531249999999999999999999999999910, 15442030112010845085854154575294808125495910644531249999999999999999999999999960),
                (50, 11, 16989717789441411514417268335819244384765624999999999999999999999999999999999998, 16989717789441411514417268335819244384765624999999999999999999999999999999999999),
                (50, 12, 906715825161540764713870551678693778322113702664995991927295792835250073659829652, 906715825161540764713870551678693778322113702664995991927295792835250073659829653),
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                (80, 19, 86271829334882047342934448278462818155638862152129831939531552797491199999999999999999999999999999999999999999999999999999999999999999999999999859, 86271829334882047342934448278462818155638862152129831939531552797491199999999999999999999999999999999999999999999999999999999999999999999999999939),
                (80, 20, 7169648795631636594807169380766160571782102481709810416532605081195359785761699436167307097884590079999999999999999999999999999999999999999999999860, 7169648795631636594807169380766160571782102481709810416532605081195359785761699436167307097884590079999999999999999999999999999999999999999999999940),
                (80, 21, 7528358180397691732123374635444051650635978522183079951251686257400687445934079999999999999999999999999999999999999999999999999999999999999999999998, 7528358180397691732123374635444051650635978522183079951251686257400687445934079999999999999999999999999999999999999999999999999999999999999999999999),
                (80, 22, 7887028195409162698095754297848609976183291119063618431405892536383568019332255598485067357994016716055346084712278578909054924946564612364497825191, 7887028195409162698095754297848609976183291119063618431405892536383568019332255598485067357994016716055346084712278578909054924946564612364497825192),
                (80, 23, 8245746323238606476073426939161185287385492298575690900842381598514230352776538527367307097884590079999999999999999999999999999999999999999999999863, 8245746323238606476073426939161185287385492298575690900842381598514230352776538527367307097884590079999999999999999999999999999999999999999999999943),
                (80, 24, 8604464441941480402729513631070699555074663455249686132035547619895009684771468728479405310371430399999999999999999999999999999999999999999999999998, 8604464441941480402729513631070699555074663455249686132035547619895009684771468728479405310371430399999999999999999999999999999999999999999999999999),
                (80, 25, 8963182560645839814649278275667045641703871683410394487016193171933062687698581347951555976069388488713071664162346166439116799999999999999999999998, 8963182560645839814649278275667045641703871683410394487016193171933062687698581347951555976069388488713071664162346166439116799999999999999999999999),
                (80, 26, 736066704035441474651755464016562111092555846876872510916086120386177480874760931319364803158153503336373944319999999999999999999999999999999999999998, 736066704035441474651755464016562111092555846876872510916086120386177480874760931319364803158153503336373944319999999999999999999999999999999999999999),
                (80, 27, 764391126388925432627810816159895846709462591273283593232076293771261305102818552605451199871942533736373944319999999999999999999999999999999999999998, 764391126388925432627810816159895846709462591273283593232076293771261305102818552605451199871942533736373944319999999999999999999999999999999999999999),
                (80, 28, 792715490465438241728467592099222855196827995555630904283054116391439593874861640908799999999999999999999999999999999999999999999999999999999999999868, 792715490465438241728467592099222855196827995555630904283054116391439593874861640908799999999999999999999999999999999999999999999999999999999999999948),
                (80, 29, 821039912089890551453691239920837552940736703514352570394763704373347768538207086942739774038260238567226736639999999999999999999999999999999999999869, 821039912089890551453691239920837552940736703514352570394763704373347768538207086942739774038260238567226736639999999999999999999999999999999999999949),
                (80, 30, 849364334443688327383589715437251625603045613334539021348085284029370116805727086942739774038260238567226736639999999999999999999999999999999999999870, 849364334443688327383589715437251625603045613334539021348085284029370116805727086942739774038260238567226736639999999999999999999999999999999999999950),
                (80, 31, 877688698510524905322094166336060687422995863859322177740272750924793487789493874693429919431608238079999999999999999999999999999999999999999999999998, 877688698510524905322094166336060687422995863859322177740272750924793487789493874693429919431608238079999999999999999999999999999999999999999999999999),
                (80, 32, 906008577023912941182140594903346566353938227682989548962092380909155458744319999999999999999999999999999999999999999999999999999999999999999999999872, 906008577023912941182140594903346566353938227682989548962092380909155458744319999999999999999999999999999999999999999999999999999999999999999999999952),
                (80, 33, 934332998657608999628321151427529183095671632158776985801135716642359818802286730318184061034037692647226736639999999999999999999999999999999999999873, 934332998657608999628321151427529183095671632158776985801135716642359818802286730318184061034037692647226736639999999999999999999999999999999999999953),
                (80, 34, 962657417228853563720539861965851210456231018281991250574673953729755955637865899296392091713076474203028328206607117904827700265822358814369967684660, 962657417228853563720539861965851210456231018281991250574673953729755955637865899296392091713076474203028328206607117904827700265822358814369967684661),
                (80, 35, 990981785078244096823798513665184128230073407138530298012312993241974413806807168566377577324950323199999999999999999999999999999999999999999999999998, 990981785078244096823798513665184128230073407138530298012312993241974413806807168566377577324950323199999999999999999999999999999999999999999999999999),
                (80, 36, 1019306207432158308189713292121084570199667787749702698455366698799009269005136387741581698965962307352773263359999999999999999999999999999999999999876, 1019306207432158308189713292121084570199667787749702698455366698799009269005136387741581698965962307352773263359999999999999999999999999999999999999956),
                (80, 37, 1047630629065854366396896870606658855712903539728913959835799645362670302178193269681815938965962307352773263359999999999999999999999999999999999999877, 1047630629065854366396896870606658855712903539728913959835799645362670302178193269681815938965962307352773263359999999999999999999999999999999999999957),
                (80, 38, 1075954993862037896503334268260034555934741770759907707606142672200445216668496387741581698965962307352773263359999999999999999999999999999999999999878, 1075954993862037896503334268260034555934741770759907707606142672200445216668496387741581698965962307352773263359999999999999999999999999999999999999958),
                (80, 39, 1104279415486490205989560937964324072392177435547261848826003875807887359999999999999999999999999999999999999999999999999999999999999999999999999999879, 1104279415486490205989560937964324072392177435547261848826003875807887359999999999999999999999999999999999999999999999999999999999999999999999999999959),
                (80, 40, 176684706477838432958329750074291851582748389687561895812160620129261977600000000000000000000000000000000000000000000000000000000000000000000000000000000, 89460610874854902763711265860400937510252349208892099145397782343930115240506329113924050632911392405063291139240506329113924050632911392405063291139240),
                (80, 41, 175580427062351942752340189136327527510356212252014633963334616253454090240000000000000000000000000000000000000000000000000000000000000000000000000000000, 91697126146726275332804047506910960948008657939114401624032726902528368121518987341772151898734177215189873417721518987341772151898734177215189873417721),
                (80, 42, 175580427062351942752340189136327527510356212252014633963334616253454090240000000000000000000000000000000000000000000000000000000000000000000000000000000, 93933641418597647901896829153420984385764966669336704102667671461126621002531645569620253164556962025316455696202531645569620253164556962025316455696202),
                (80, 43, 175580427062351942752340189136327527510356212252014633963334616253454090240000000000000000000000000000000000000000000000000000000000000000000000000000000, 96170156690469020470989610799931007823521275399559006581302616019724873883544303797468354430379746835443037974683544303797468354430379746835443037974683),
                (80, 44, 175580427062351942752340189136327527510356212252014633963334616253454090240000000000000000000000000000000000000000000000000000000000000000000000000000000, 98406671962340393040082392446441031261277584129781309059937560578323126764556962025316455696202531645569620253164556962025316455696202531645569620253164),
                (80, 45, 175580427062351942752340189136327527510356212252014633963334616253454090240000000000000000000000000000000000000000000000000000000000000000000000000000000, 100643187234211765609175174092951054699033892860003611538572505136921379645569620253164556962025316455696202531645569620253164556962025316455696202531645),
                (80, 46, 175580427062351942752340189136327527510356212252014633963334616253454090240000000000000000000000000000000000000000000000000000000000000000000000000000000, 102879702506083138178267955739461078136790201590225914017207449695519632526582278481012658227848101265822784810126582278481012658227848101265822784810126),
                (80, 47, 175580427062351942752340189136327527510356212252014633963334616253454090240000000000000000000000000000000000000000000000000000000000000000000000000000000, 105116217777954510747360737385971101574546510320448216495842394254117885407594936708860759493670886075949367088607594936708860759493670886075949367088607),
                (80, 48, 175580427062351942752340189136327527510356212252014633963334616253454090240000000000000000000000000000000000000000000000000000000000000000000000000000000, 107352733049825883316453519032481125012302819050670518974477338812716138288607594936708860759493670886075949367088607594936708860759493670886075949367088),
                (80, 49, 175580427062351942752340189136327527510356212252014633963334616253454090240000000000000000000000000000000000000000000000000000000000000000000000000000000, 109589248321697255885546300678991148450059127780892821453112283371314391169620253164556962025316455696202531645569620253164556962025316455696202531645569),
                (80, 50, 175580427062351942752340189136327527510356212252014633963334616253454090240000000000000000000000000000000000000000000000000000000000000000000000000000000, 111825763593568628454639082325501171887815436511115123931747227929912644050632911392405063291139240506329113924050632911392405063291139240506329113924050),
                (80, 51, 175580427062351942752340189136327527510356212252014633963334616253454090240000000000000000000000000000000000000000000000000000000000000000000000000000000, 114062278865440001023731863972011195325571745241337426410382172488510896931645569620253164556962025316455696202531645569620253164556962025316455696202531),
                (80, 52, 175580427062351942752340189136327527510356212252014633963334616253454090240000000000000000000000000000000000000000000000000000000000000000000000000000000, 116298794137311373592824645618521218763328053971559728889017117047109149812658227848101265822784810126582278481012658227848101265822784810126582278481012),
                (80, 53, 175580427062351942752340189136327527510356212252014633963334616253454090240000000000000000000000000000000000000000000000000000000000000000000000000000000, 118535309409182746161917427265031242201084362701782031367652061605707402693670886075949367088607594936708860759493670886075949367088607594936708860759493),
                (80, 54, 175571195976613110373274495150361731988813558893764356071317086377304758681600000000000000000000000000000000000000000000000000000000000000000000000000000, 120771824681054118731010208911541265638840671432004333846287006164305655574683544303797468354430379746835443037974683544303797468354430379746835443037974),
                (80, 55, 175571195976613110373274495150361731988813558893764356071317086377304758681600000000000000000000000000000000000000000000000000000000000000000000000000000, 123008339952925491300102990558051289076596980162226636324921950722903908455696202531645569620253164556962025316455696202531645569620253164556962025316455),
                (80, 56, 175571195976613110373274495150361731988813558893764356071317086377304758681600000000000000000000000000000000000000000000000000000000000000000000000000000, 125244855224796863869195772204561312514353288892448938803556895281502161336708860759493670886075949367088607594936708860759493670886075949367088607594936),
                (80, 57, 175571195976613110373274495150361731988813558893764356071317086377304758681600000000000000000000000000000000000000000000000000000000000000000000000000000, 127481370496668236438288553851071335952109597622671241282191839840100414217721518987341772151898734177215189873417721518987341772151898734177215189873417),
                (80, 58, 175571195976613110373274495150361731988813558893764356071317086377304758681600000000000000000000000000000000000000000000000000000000000000000000000000000, 129717885768539609007381335497581359389865906352893543760826784398698667098734177215189873417721518987341772151898734177215189873417721518987341772151898),
                (80, 59, 175571195976613110373274495150361731988813558893764356071317086377304758681600000000000000000000000000000000000000000000000000000000000000000000000000000, 131954401040410981576474117144091382827622215083115846239461728957296919979746835443037974683544303797468354430379746835443037974683544303797468354430379),
                (80, 60, 175571195976613110373274495150361731988813558893764356071317086377304758681600000000000000000000000000000000000000000000000000000000000000000000000000000, 134190916312282354145566898790601406265378523813338148718096673515895172860759493670886075949367088607594936708860759493670886075949367088607594936708860),
                (80, 61, 175571195976613110373274495150361731988813558893764356071317086377304758681600000000000000000000000000000000000000000000000000000000000000000000000000000, 136427431584153726714659680437111429703134832543560451196731618074493425741772151898734177215189873417721518987341772151898734177215189873417721518987341),
                (80, 62, 175571195976613110373274495150361731988813558893764356071317086377304758681600000000000000000000000000000000000000000000000000000000000000000000000000000, 138663946856025099283752462083621453140891141273782753675366562633091678622784810126582278481012658227848101265822784810126582278481012658227848101265822),
                (80, 63, 175571195976613110373274495150361731988813558893764356071317086377304758681600000000000000000000000000000000000000000000000000000000000000000000000000000, 140900462127896471852845243730131476578647450004005056154001507191689931503797468354430379746835443037974683544303797468354430379746835443037974683544303),
                (80, 64, 175571195976613110373274495150361731988813558893764356071317086377304758681600000000000000000000000000000000000000000000000000000000000000000000000000000, 143136977399767844421938025376641500016403758734227358632636451750288184384810126582278481012658227848101265822784810126582278481012658227848101265822784),
                (80, 65, 175571195113894817024454021721017249204185377337375734550018766981989230706688000000000000000000000000000000000000000000000000000000000000000000000000000, 145373492671639216991030807023151523454160067464449661111271396308886437265822784810126582278481012658227848101265822784810126582278481012658227848101265),
                (80, 66, 175571195113894817024454021721017249204185377337375734550018766981989230706688000000000000000000000000000000000000000000000000000000000000000000000000000, 147610007943510589560123588669661546891916376194671963589906340867484690146835443037974683544303797468354430379746835443037974683544303797468354430379746),
                (80, 67, 175571195104843225555541531524331194724773903679566701100341570869681202187468800000000000000000000000000000000000000000000000000000000000000000000000000, 149846523215381962129216370316171570329672684924894266068541285426082943027848101265822784810126582278481012658227848101265822784810126582278481012658227),
                (80, 68, 175571195104843225555541531524331194724773903679566701100341570869681202187468800000000000000000000000000000000000000000000000000000000000000000000000000, 152083038487253334698309151962681593767428993655116568547176229984681195908860759493670886075949367088607594936708860759493670886075949367088607594936708),
                (80, 69, 175571195104843225555541531524331194724773903679566701100341570869681202187468800000000000000000000000000000000000000000000000000000000000000000000000000, 154319553759124707267401933609191617205185302385338871025811174543279448789873417721518987341772151898734177215189873417721518987341772151898734177215189),
                (80, 70, 175571195104843225555541531524331194724773903679566701100341570869681202187468800000000000000000000000000000000000000000000000000000000000000000000000000, 156556069030996079836494715255701640642941611115561173504446119101877701670886075949367088607594936708860759493670886075949367088607594936708860759493670),
                (80, 71, 175571195104843225555541531524331194724773903679566701100341570869681202187468800000000000000000000000000000000000000000000000000000000000000000000000000, 158792584302867452405587496902211664080697919845783475983081063660475954551898734177215189873417721518987341772151898734177215189873417721518987341772151),
                (80, 72, 175571195104843213839549083397344174581393323871231105623262207558592575145372549120000000000000000000000000000000000000000000000000000000000000000000000, 161029099574738824974680278548721687518454228576005778461716008219074207432911392405063291139240506329113924050632911392405063291139240506329113924050632),
                (80, 73, 175571195104843213838040636862208762157819107640578567202908816185817931617828910556774400000000000000000000000000000000000000000000000000000000000000000, 163265614846610197543773060195231710956210537306228080940350952777672460313924050632911392405063291139240506329113924050632911392405063291139240506329113),
                (80, 74, 175571195104843213838040636862208762157819107640578567202908816185817931617828910556774400000000000000000000000000000000000000000000000000000000000000000, 165502130118481570112865841841741734393966846036450383418985897336270713194936708860759493670886075949367088607594936708860759493670886075949367088607594),
                (80, 75, 175571195104843213838040636862208762157819107640578567202908816185817931617828910556774400000000000000000000000000000000000000000000000000000000000000000, 167738645390352942681958623488251757831723154766672685897620841894868966075949367088607594936708860759493670886075949367088607594936708860759493670886075),
                (80, 76, 175571195104843213838040636862205056804280651806007052323399138647856463947729904201151545344000000000000000000000000000000000000000000000000000000000000, 169975160662224315251051405134761781269479463496894988376255786453467218956962025316455696202531645569620253164556962025316455696202531645569620253164556),
                (80, 77, 175571195104843213838040636862205056804207504201479193037011460487569338336526152588352194996695662523652306370560000000000000000000000000000000000000000, 172211675934095687820144186781271804707235772227117290854890731012065471837974683544303797468354430379746835443037974683544303797468354430379746835443037),
                (80, 78, 175571195104843213838040636862205056804207504201479193037011460487569338336526152588352194996695662523652306370560000000000000000000000000000000000000000, 174448191205967060389236968427781828144992080957339593333525675570663724718987341772151898734177215189873417721518987341772151898734177215189873417721518),
                (80, 79, 176684706477838432958329750074291851582748389687561895812160620129261977600000000000000000000000000000000000000000000000000000000000000000000000000000000, 176684706477838432958329750074291851582748389687561895812160620129261977599999999999999999999999999999999999999999999999999999999999999999999999999999999),
                ]



"""### `table_bases` for bases 80-100"""

table_bases += [
    (81, 0, 3131879606492502363897623259852309766515790308883390627397052049901151154569228223107019286300551283158059404119309880632990446156639030701056269696057118332, 3131879606492502363897623259852309766515790308883390627397052049901151154569228223107019286300551283158059404119309880632990446156639030701056269696057118333),
(81, 1, 3495031234939188300090902828051863362772743810135426776882042385735348864709777, 3495031234939188300090902828051863362772743810135426776882042385735348864709858),
(81, 2, 34301433818510654479997622149056072840806381240712147713459454414705167747070180824150668119248233312003, 34301433818510654479997622149056072840806381240712147713459454414705167747070180824150668119248233312084),
(81, 3, 1041896496520091376929393362139841573971575035939107433387648106094144679296649787641629987098728164345595245179443045, 1041896496520091376929393362139841573971575035939107433387648106094144679296649787641629987098728164345595245179443126),
(81, 4, 56121880626308179125471268779897292988672593759335333117796879976563280676173844919360793278783332500288629036346934921791950, 56121880626308179125471268779897292988672593759335333117796879976563280676173844919360793278783332500288629036346934921792031),
(81, 5, 35634504986367753906117248446358993264100595335686175527572396905419086695511503135111814071354648972407775206844087979175363302409, 35634504986367753906117248446358993264100595335686175527572396905419086695511503135111814071354648972407775206844087979175363302490),
(81, 6, 291188638680572229521718228626871800329226329476334632108359879909707279856259030938121083905334089229732596174992215951574725253153543, 272539727165403186690049605130508559010929615726177848230513411466629510950399747224999601834722824554636168071060775949515947011392620),
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]

"""### Verifying and extending `table_bases`"""

if not "SKIP":
  for base, digit, eq, gt in tqdm(table_bases):
    if base == 1:
      pass
    elif base == 2:
      if eq is not None:
        for smaller in range(2, eq):
          assert slow_count_matches_to_k(digit, smaller, base) < smaller
        assert slow_count_matches_to_k(digit, eq, base) == eq
      if gt is not None:
        for smaller in range(2, gt):
          assert slow_count_matches_to_k(digit, smaller, base) <= smaller
        assert slow_count_matches_to_k(digit, gt, base) > gt
    else:
      vgt, veq = unbounded_binary_ge_search(digit, base)
      assert vgt == gt, f"{base=} {digit=} {vgt=} {gt=}"
      assert veq == eq, f"{base=} {digit=} {veq=} {eq=}"

# Extend the table above:
if not "SKIP":
  for base in range(101, 201):
    for digit in range(base):
      gt, eq = unbounded_binary_ge_search(digit, base, tqdm_after=60*60)
      print(f"({base}, {digit}, {eq}, {gt}),")

"""### Missing values in `table_bases`"""

[(base, digit) for base, digit, eq, gt in table_bases if eq is None]

"""Proposed sequence https://oeis.org/draft/A364972:"""

print([base for base, digit, eq, gt in table_bases if base>1 and eq is None])

# And when it's well-defined, what is it?
for base, digit, eq, gt in table_bases:
  if digit == 0 and base>1 and eq is not None:
    print(f"{base}\t{eq}\t{gt}\t{letterbase(eq, base)}\t{letterbase(gt, base)}")

"""### Verify https://oeis.org/A092175"""

A092175 = [2, 3, 13, 29, 182, 427, 3931, 8185, 102781, 199991, 3179143, 5971957, 114818731, 210826995, 4754446861, 8589934577, 222195898594, 396718580719, 11575488191148, 20479999999981, 665306762187614, 1168636602822635]
ours = [gt for base, digit, eq, gt in table_bases if base>1 and digit == 1]
assert A092175[1:] == ours[:len(A092175)-1], f"\n{A092175[1:]}\n{ours[:len(A092175)-1]}"
#ours
for base, digit, eq, gt in table_bases:
  if base>1 and digit==1:
    print(f"{base} {gt}")
print("")

"""### `table_all_equal_by_base`

"""

# Verify https://oeis.org/A165617
A165617 = [2, 4, 8, 4, 21, 5, 45, 49, 83]
# a(n) is the number of positive integers k such that k is equal to the number of 1's in the digits of the base-n expansion of all positive integers <= k.
# Example: a(3)=4 since there are four values of k such that k is equal to the number of 1's in the digits of the base-3 expansion of all numbers <= k, namely, 1, 4, 5, 12.

alleq, _, _ = multidigit_all_eq_search(1, base=3, safeleft=0, sizebound=100)
assert [1, 4, 5, 12] == alleq, alleq

for base in range(2,len(A165617)+2):
  alleq, _, _ = multidigit_all_eq_search(1, base, safeleft=0, sizebound=100)
  assert A165617[base-2] == len(alleq), f"{base=} {A165617[base-2]=} {alleq=}"

# Extend the A165617 sequence
if not "ALL BASES AND DIGITS rather than just the sequence in question":
  # Populate table_all_equal_by_base below
  for base in range(42, 100):
    for digit in range(base):
      alleq = find_all_equal(digit, base)
      if len(alleq) < 10:
        print(f"({base}, {digit}, {len(alleq)}, {alleq}),")
      else:
        print(f"({base}, {digit}, {len(alleq)}, None),")  # Do not print the actual list, just its length.

# Commented out IPython magic to ensure Python compatibility.
# %%html
# <style>
# div.output_area pre {
#     white-space: pre, nowrap;
# }
# div.code pre {
#     white-space: nowrap;
# }
# </style>
#

"""#### from 1-9"""

table_all_equal_by_base = [
(2, 0, 1, [8]),
(2, 1, 2, [1, 2]),
(3, 0, 0, []),
(3, 1, 4, [1, 4, 5, 12]),
(3, 2, 1, [27]),
(4, 0, 0, []),
(4, 1, 8, [1, 25, 26, 27, 28, 32, 33, 84]),
(4, 2, 1, [256]),
(4, 3, 1, [256]),
(5, 0, 0, []),
(5, 1, 4, [1, 181, 338, 780]),
(5, 2, 1, [3125]),
(5, 3, 6, [2787, 2788, 2793, 2794, 2940, 3125]),
(5, 4, 1, [3125]),
(6, 0, 0, []),
(6, 1, 21, [1, 421, 422, 423, 424, 425, 426, 432, 433, 1726, 3888, 3889, 4309, 4310, 4311, 4312, 4313, 4314, 4320, 4321, 9330]),
(6, 2, 8, [3878, 3879, 3880, 3881, 3882, 3883, 3888, 46656]),
(6, 3, 1, [46656]),
(6, 4, 7, [42768, 44921, 44922, 44923, 44924, 44925, 46656]),
(6, 5, 1, [46656]),
(7, 0, 0, []),
(7, 1, 5, [1, 3930, 6044, 61879, 137256]),
(7, 2, 3, [46340, 61814, 823543]),
(7, 3, 1, [823543]),
(7, 4, 3, [761729, 761730, 823543]),
(7, 5, 7, [761664, 817499, 817500, 817507, 817508, 819609, 823543]),
(7, 6, 1, [823543]),
(8, 0, 0, []),
(8, 1, 45, [1, 8177, 8178, 8179, 8180, 8181, 8182, 8183, 8184, 8192, 8193, 49137, 49138, 49139, 49140, 49141, 49142, 49143, 49144, 90112, 90113, 322096, 1048576, 1048577, 1056753, 1056754, 1056755, 1056756, 1056757, 1056758, 1056759, 1056760, 1056768, 1056769, 1097713, 1097714, 1097715, 1097716, 1097717, 1097718, 1097719, 1097720, 1138688, 1138689, 2396744]),
(8, 2, 10, [679922, 679923, 679924, 679925, 679926, 679927, 679928, 679929, 1048576, 16777216]),
(8, 3, 10, [1048563, 1048564, 1048565, 1048566, 1048567, 1048568, 1048569, 1048570, 1048576, 16777216]),
(8, 4, 1, [16777216]),
(8, 5, 10, [15728640, 16097269, 16097270, 16097271, 16097272, 16097273, 16097274, 16097275, 16097276, 16777216]),
(8, 6, 19, [15638528, 15679479, 15679480, 15679481, 15679482, 15679483, 15679484, 15679485, 15728640, 16455117, 16687104, 16728055, 16728056, 16728057, 16728058, 16728059, 16728060, 16728061, 16777216]),
(8, 7, 1, [16777216]),
(9, 0, 0, []),
(9, 1, 49, [1, 102772, 102773, 102774, 102775, 102776, 102777, 102778, 102779, 102780, 142155, 142156, 737002, 737003, 737004, 737005, 737006, 737007, 737008, 737009, 737010, 1594323, 1594324, 1697095, 1697096, 1697097, 1697098, 1697099, 1697100, 1697101, 1697102, 1697103, 1736478, 1736479, 5732125, 22320522, 22320523, 22423294, 22423295, 22423296, 22423297, 22423298, 22423299, 22423300, 22423301, 22423302, 22462677, 22462678, 48427560]),
(9, 2, 13, [1594307, 1594308, 1594309, 1594310, 1594311, 1594312, 1594313, 1594314, 1594315, 1594323, 11897278, 22320522, 387420489]),
(9, 3, 2, [22320522, 387420489]),
(9, 4, 1, [387420489]),
(9, 5, 12, [365099967, 366694277, 366694278, 366694279, 366694280, 366694281, 366694282, 366694283, 366694284, 366694285, 366694290, 387420489]),
(9, 6, 11, [365099967, 375523197, 375523198, 375523199, 375523200, 375523201, 375523202, 375523203, 375523204, 375523205, 387420489]),
(9, 7, 39, [364957812, 364997168, 364997169, 364997170, 364997171, 364997172, 364997173, 364997174, 364997175, 365099967, 381688360, 385684011, 385723367, 385723368, 385723369, 385723370, 385723371, 385723372, 385723373, 385723374, 385826166, 386683460, 386683461, 386683462, 386683463, 386683464, 386683465, 386683466, 386683467, 387278334, 387317690, 387317691, 387317692, 387317693, 387317694, 387317695, 387317696, 387317697, 387420489]),
(9, 8, 1, [387420489]),
(10, 0, 0, []),
(10, 1, 83, [1, 199981, 199982, 199983, 199984, 199985, 199986, 199987, 199988, 199989, 199990, 200000, 200001, 1599981, 1599982, 1599983, 1599984, 1599985, 1599986, 1599987, 1599988, 1599989, 1599990, 2600000, 2600001, 13199998, 35000000, 35000001, 35199981, 35199982, 35199983, 35199984, 35199985, 35199986, 35199987, 35199988, 35199989, 35199990, 35200000, 35200001, 117463825, 500000000, 500000001, 500199981, 500199982, 500199983, 500199984, 500199985, 500199986, 500199987, 500199988, 500199989, 500199990, 500200000, 500200001, 501599981, 501599982, 501599983, 501599984, 501599985, 501599986, 501599987, 501599988, 501599989, 501599990, 502600000, 502600001, 513199998, 535000000, 535000001, 535199981, 535199982, 535199983, 535199984, 535199985, 535199986, 535199987, 535199988, 535199989, 535199990, 535200000, 535200001, 1111111110]),
(10, 2, 7, [28263827, 35000000, 242463827, 500000000, 528263827, 535000000, 10000000000]),
(10, 3, 12, [371599983, 371599984, 371599985, 371599986, 371599987, 371599988, 371599989, 371599990, 371599991, 371599992, 500000000, 10000000000]),
(10, 4, 12, [499999984, 499999985, 499999986, 499999987, 499999988, 499999989, 499999990, 499999991, 499999992, 499999993, 500000000, 10000000000]),
(10, 5, 1, [10000000000]),
(10, 6, 12, [9500000000, 9628399986, 9628399987, 9628399988, 9628399989, 9628399990, 9628399991, 9628399992, 9628399993, 9628399994, 9628399995, 10000000000]),
(10, 7, 7, [9465000000, 9471736170, 9500000000, 9757536170, 9965000000, 9971736170, 10000000000]),
(10, 8, 43, [9465000000, 9486799989, 9486799990, 9486799991, 9486799992, 9486799993, 9486799994, 9486799995, 9486799996, 9486799997, 9497400000, 9498399989, 9498399990, 9498399991, 9498399992, 9498399993, 9498399994, 9498399995, 9498399996, 9498399997, 9500000000, 9882536171, 9965000000, 9986799989, 9986799990, 9986799991, 9986799992, 9986799993, 9986799994, 9986799995, 9986799996, 9986799997, 9997400000, 9998399989, 9998399990, 9998399991, 9998399992, 9998399993, 9998399994, 9998399995, 9998399996, 9998399997, 10000000000]),
]

"""#### 10-14"""

table_all_equal_by_base += [
    (10, 9, 1, [10000000000]),
(11, 0, 1, [3152738985031]),
(11, 1, 10, [1, 3179142, 4141637, 29027978, 54643915, 275032201, 806958983, 2726979313, 13399512860, 28531167060]),
(11, 2, 4, [581404546, 806958838, 13399512860, 285311670611]),
(11, 3, 4, [13344870724, 13351893425, 13399512715, 285311670611]),
(11, 4, 6, [11484304153, 12592555801, 12676291750, 13344870724, 13399488902, 285311670611]),
(11, 5, 1, [285311670611]),
(11, 6, 6, [271912181709, 271966799887, 272635378857, 272719114810, 273827366455, 285311670611]),
(11, 7, 4, [271912157896, 271959777183, 271966799887, 285311670611]),
(11, 8, 5, [271912157751, 284504711773, 284504711774, 284730266062, 285311670611]),
(11, 9, 8, [271912157751, 282584691295, 284504711628, 285036638406, 285257026696, 285282642629, 285307528974, 285311670611]),
(11, 10, 1, [285311670611]),
(12, 0, 0, []),
(12, 1, 269, [1, 5971945, 5971946, 5971947, 5971948, 5971949, 5971950, 5971951, 5971952, 5971953, 5971954, 5971955, 5971956, 5971968, 5971969, 59719657, 59719658, 59719659, 59719660, 59719661, 59719662, 59719663, 59719664, 59719665, 59719666, 59719667, 59719668, 89579520, 89579521, 612997630, 1289945088, 1289945089, 1295917033, 1295917034, 1295917035, 1295917036, 1295917037, 1295917038, 1295917039, 1295917040, 1295917041, 1295917042, 1295917043, 1295917044, 1295917056, 1295917057, 1349664745, 1349664746, 1349664747, 1349664748, 1349664749, 1349664750, 1349664751, 1349664752, 1349664753, 1349664754, 1349664755, 1349664756, 1379524608, 1379524609, 6449725417, 6449725418, 6449725419, 6449725420, 6449725421, 6449725422, 6449725423, 6449725424, 6449725425, 6449725426, 6449725427, 6449725428, 20639121408, 20639121409, 20645093353, 20645093354, 20645093355, 20645093356, 20645093357, 20645093358, 20645093359, 20645093360, 20645093361, 20645093362, 20645093363, 20645093364, 20645093376, 20645093377, 20698841065, 20698841066, 20698841067, 20698841068, 20698841069, 20698841070, 20698841071, 20698841072, 20698841073, 20698841074, 20698841075, 20698841076, 20728700928, 20728700929, 21252119038, 21929066496, 21929066497, 21935038441, 21935038442, 21935038443, 21935038444, 21935038445, 21935038446, 21935038447, 21935038448, 21935038449, 21935038450, 21935038451, 21935038452, 21935038464, 21935038465, 21988786153, 21988786154, 21988786155, 21988786156, 21988786157, 21988786158, 21988786159, 21988786160, 21988786161, 21988786162, 21988786163, 21988786164, 22018646016, 22018646017, 70674197758, 371504185344, 371504185345, 371510157289, 371510157290, 371510157291, 371510157292, 371510157293, 371510157294, 371510157295, 371510157296, 371510157297, 371510157298, 371510157299, 371510157300, 371510157312, 371510157313, 371563905001, 371563905002, 371563905003, 371563905004, 371563905005, 371563905006, 371563905007, 371563905008, 371563905009, 371563905010, 371563905011, 371563905012, 371593764864, 371593764865, 372117182974, 372794130432, 372794130433, 372800102377, 372800102378, 372800102379, 372800102380, 372800102381, 372800102382, 372800102383, 372800102384, 372800102385, 372800102386, 372800102387, 372800102388, 372800102400, 372800102401, 372853850089, 372853850090, 372853850091, 372853850092, 372853850093, 372853850094, 372853850095, 372853850096, 372853850097, 372853850098, 372853850099, 372853850100, 372883709952, 372883709953, 377953910761, 377953910762, 377953910763, 377953910764, 377953910765, 377953910766, 377953910767, 377953910768, 377953910769, 377953910770, 377953910771, 377953910772, 392143306752, 392143306753, 392149278697, 392149278698, 392149278699, 392149278700, 392149278701, 392149278702, 392149278703, 392149278704, 392149278705, 392149278706, 392149278707, 392149278708, 392149278720, 392149278721, 392203026409, 392203026410, 392203026411, 392203026412, 392203026413, 392203026414, 392203026415, 392203026416, 392203026417, 392203026418, 392203026419, 392203026420, 392232886272, 392232886273, 392756304382, 393433251840, 393433251841, 393439223785, 393439223786, 393439223787, 393439223788, 393439223789, 393439223790, 393439223791, 393439223792, 393439223793, 393439223794, 393439223795, 393439223796, 393439223808, 393439223809, 393492971497, 393492971498, 393492971499, 393492971500, 393492971501, 393492971502, 393492971503, 393492971504, 393492971505, 393492971506, 393492971507, 393492971508, 393522831360, 393522831361, 810554586204]),
(12, 2, 92, [1289945066, 1289945067, 1289945068, 1289945069, 1289945070, 1289945071, 1289945072, 1289945073, 1289945074, 1289945075, 1289945076, 1289945077, 1289945088, 13514563562, 13514563563, 13514563564, 13514563565, 13514563566, 13514563567, 13514563568, 13514563569, 13514563570, 13514563571, 13514563572, 13514563573, 20639121408, 21929066474, 21929066475, 21929066476, 21929066477, 21929066478, 21929066479, 21929066480, 21929066481, 21929066482, 21929066483, 21929066484, 21929066485, 21929066496, 144473849834, 144473849835, 144473849836, 144473849837, 144473849838, 144473849839, 144473849840, 144473849841, 144473849842, 144473849843, 144473849844, 144473849845, 371504185344, 372794130410, 372794130411, 372794130412, 372794130413, 372794130414, 372794130415, 372794130416, 372794130417, 372794130418, 372794130419, 372794130420, 372794130421, 372794130432, 385018748906, 385018748907, 385018748908, 385018748909, 385018748910, 385018748911, 385018748912, 385018748913, 385018748914, 385018748915, 385018748916, 385018748917, 392143306752, 393433251818, 393433251819, 393433251820, 393433251821, 393433251822, 393433251823, 393433251824, 393433251825, 393433251826, 393433251827, 393433251828, 393433251829, 393433251840, 8916100448256]),
(12, 3, 40, [20639121387, 20639121388, 20639121389, 20639121390, 20639121391, 20639121392, 20639121393, 20639121394, 20639121395, 20639121396, 20639121397, 20639121398, 20639121408, 219905777643, 219905777644, 219905777645, 219905777646, 219905777647, 219905777648, 219905777649, 219905777650, 219905777651, 219905777652, 219905777653, 219905777654, 371504185344, 392143306731, 392143306732, 392143306733, 392143306734, 392143306735, 392143306736, 392143306737, 392143306738, 392143306739, 392143306740, 392143306741, 392143306742, 392143306752, 8916100448256]),
(12, 4, 3, [295916986366, 371504185344, 8916100448256]),
(12, 5, 14, [371504185325, 371504185326, 371504185327, 371504185328, 371504185329, 371504185330, 371504185331, 371504185332, 371504185333, 371504185334, 371504185335, 371504185336, 371504185344, 8916100448256]),
(12, 6, 1, [8916100448256]),
(12, 7, 3, [8544596262912, 8620183461886, 8916100448256]),
(12, 8, 14, [8544596262912, 8696194670576, 8696194670577, 8696194670578, 8696194670579, 8696194670580, 8696194670581, 8696194670582, 8696194670583, 8696194670584, 8696194670585, 8696194670586, 8696194670587, 8916100448256]),
(12, 9, 18, [8523957141504, 8531081699326, 8544596262912, 8771626598385, 8771626598386, 8771626598387, 8771626598388, 8771626598389, 8771626598390, 8771626598391, 8771626598392, 8771626598393, 8771626598394, 8771626598395, 8771626598396, 8895461326848, 8902585884670, 8916100448256]),
(12, 10, 131, [8522577616896, 8522607476723, 8522607476724, 8522607476725, 8522607476726, 8522607476727, 8522607476728, 8522607476729, 8522607476730, 8522607476731, 8522607476732, 8522607476733, 8522667196416, 8523344143870, 8523867561984, 8523897421811, 8523897421812, 8523897421813, 8523897421814, 8523897421815, 8523897421816, 8523897421817, 8523897421818, 8523897421819, 8523897421820, 8523897421821, 8523957141504, 8538146537459, 8538146537460, 8538146537461, 8538146537462, 8538146537463, 8538146537464, 8538146537465, 8538146537466, 8538146537467, 8538146537468, 8538146537469, 8543216738304, 8543246598131, 8543246598132, 8543246598133, 8543246598134, 8543246598135, 8543246598136, 8543246598137, 8543246598138, 8543246598139, 8543246598140, 8543246598141, 8543306317824, 8543983265278, 8544506683392, 8544536543219, 8544536543220, 8544536543221, 8544536543222, 8544536543223, 8544536543224, 8544536543225, 8544536543226, 8544536543227, 8544536543228, 8544536543229, 8544596262912, 8845426250494, 8894081802240, 8894111662067, 8894111662068, 8894111662069, 8894111662070, 8894111662071, 8894111662072, 8894111662073, 8894111662074, 8894111662075, 8894111662076, 8894111662077, 8894171381760, 8894848329214, 8895371747328, 8895401607155, 8895401607156, 8895401607157, 8895401607158, 8895401607159, 8895401607160, 8895401607161, 8895401607162, 8895401607163, 8895401607164, 8895401607165, 8895461326848, 8909650722803, 8909650722804, 8909650722805, 8909650722806, 8909650722807, 8909650722808, 8909650722809, 8909650722810, 8909650722811, 8909650722812, 8909650722813, 8914720923648, 8914750783475, 8914750783476, 8914750783477, 8914750783478, 8914750783479, 8914750783480, 8914750783481, 8914750783482, 8914750783483, 8914750783484, 8914750783485, 8914810503168, 8915487450622, 8916010868736, 8916040728563, 8916040728564, 8916040728565, 8916040728566, 8916040728567, 8916040728568, 8916040728569, 8916040728570, 8916040728571, 8916040728572, 8916040728573, 8916100448256]),
(12, 11, 1, [8916100448256]),
(13, 0, 1, [3950024143546664]),
(13, 1, 11, [1, 143610511, 1275200603, 2178707384, 14564076747, 36170333076, 168270263261, 621452898464, 2023925853448, 11959840815622, 25239592216020]),
(13, 2, 10, [30411771424, 36170293121, 36170293122, 351946494373, 621452278516, 621452581922, 621452898464, 4119784765135, 11959840815622, 302875106592253]),
(13, 3, 6, [535464000311, 621452278501, 621452278502, 6264577301480, 11959840812834, 302875106592253]),
(13, 4, 3, [11959840812819, 11959840812820, 302875106592253]),
(13, 5, 10, [10568722104311, 11338387917370, 11374414642723, 11472763746671, 11959697205126, 11959772944331, 11959840195871, 11959840331161, 11959840812819, 302875106592253]),
(13, 6, 1, [302875106592253]),
(13, 7, 12, [290915265779434, 290915266261088, 290915266396382, 290915333647919, 290915409387127, 291402342845579, 291500691949530, 291509760265547, 291536718674883, 291536718674884, 292306384487938, 302875106592253]),
(13, 8, 2, [290915265779434, 302875106592253]),
(13, 9, 6, [290915265779419, 290915265779420, 296610529290769, 302253654313752, 302339642591938, 302875106592253]),
(13, 10, 10, [290915265776631, 298755321827115, 302253653693789, 302253654010327, 302253654313737, 302253654313738, 302523160097876, 302838936299132, 302844694820825, 302875106592253]),
(13, 11, 10, [290915265776631, 300851180738802, 302253653693789, 302706836328988, 302838936259177, 302872927884869, 302873831391646, 302874962981742, 302874991773519, 302875106592253]),
(13, 12, 1, [302875106592253]),
(14, 0, 0, []),
(14, 1, 202, [1, 210826981, 210826982, 210826983, 210826984, 210826985, 210826986, 210826987, 210826988, 210826989, 210826990, 210826991, 210826992, 210826993, 210826994, 210827008, 210827009, 2529924069, 2529924070, 2529924071, 2529924072, 2529924073, 2529924074, 2529924075, 2529924076, 2529924077, 2529924078, 2529924079, 2529924080, 2529924081, 2529924082, 3584059136, 3584059137, 30991570149, 30991570150, 30991570151, 30991570152, 30991570153, 30991570154, 30991570155, 30991570156, 30991570157, 30991570158, 30991570159, 30991570160, 30991570161, 30991570162, 58609908224, 58609908225, 384759289573, 384759289574, 384759289575, 384759289576, 384759289577, 384759289578, 384759289579, 384759289580, 384759289581, 384759289582, 384759289583, 384759289584, 384759289585, 384759289586, 1043382862592, 1043382862593, 1043593689573, 1043593689574, 1043593689575, 1043593689576, 1043593689577, 1043593689578, 1043593689579, 1043593689580, 1043593689581, 1043593689582, 1043593689583, 1043593689584, 1043593689585, 1043593689586, 1043593689600, 1043593689601, 4852605243134, 19266426126080, 19266426126081, 19266636953061, 19266636953062, 19266636953063, 19266636953064, 19266636953065, 19266636953066, 19266636953067, 19266636953068, 19266636953069, 19266636953070, 19266636953071, 19266636953072, 19266636953073, 19266636953074, 19266636953088, 19266636953089, 396857386627072, 396857386627073, 396857597454053, 396857597454054, 396857597454055, 396857597454056, 396857597454057, 396857597454058, 396857597454059, 396857597454060, 396857597454061, 396857597454062, 396857597454063, 396857597454064, 396857597454065, 396857597454066, 396857597454080, 396857597454081, 396859916551141, 396859916551142, 396859916551143, 396859916551144, 396859916551145, 396859916551146, 396859916551147, 396859916551148, 396859916551149, 396859916551150, 396859916551151, 396859916551152, 396859916551153, 396859916551154, 396860970686208, 396860970686209, 396888378197221, 396888378197222, 396888378197223, 396888378197224, 396888378197225, 396888378197226, 396888378197227, 396888378197228, 396888378197229, 396888378197230, 396888378197231, 396888378197232, 396888378197233, 396888378197234, 396915996535296, 396915996535297, 397242145916645, 397242145916646, 397242145916647, 397242145916648, 397242145916649, 397242145916650, 397242145916651, 397242145916652, 397242145916653, 397242145916654, 397242145916655, 397242145916656, 397242145916657, 397242145916658, 397900769489664, 397900769489665, 397900980316645, 397900980316646, 397900980316647, 397900980316648, 397900980316649, 397900980316650, 397900980316651, 397900980316652, 397900980316653, 397900980316654, 397900980316655, 397900980316656, 397900980316657, 397900980316658, 397900980316672, 397900980316673, 401709991870206, 416123812753152, 416123812753153, 416124023580133, 416124023580134, 416124023580135, 416124023580136, 416124023580137, 416124023580138, 416124023580139, 416124023580140, 416124023580141, 416124023580142, 416124023580143, 416124023580144, 416124023580145, 416124023580146, 416124023580160, 416124023580161, 854769755812154]),
(14, 2, 37, [803108615022, 1043382862592, 10123912924134, 10123912924135, 10123912924136, 10123912924137, 10123912924138, 10123912924139, 10123912924140, 10123912924141, 10123912924142, 10123912924143, 10123912924144, 10123912924145, 10123912924146, 10123912924147, 19266426126080, 128768076638206, 396857386627072, 397660495242094, 397900769489664, 406981299551206, 406981299551207, 406981299551208, 406981299551209, 406981299551210, 406981299551211, 406981299551212, 406981299551213, 406981299551214, 406981299551215, 406981299551216, 406981299551217, 406981299551218, 406981299551219, 416123812753152, 11112006825558016]),
(14, 3, 7, [15393391704939, 19266426126080, 195599041754987, 396857386627072, 412250778332011, 416123812753152, 11112006825558016]),
(14, 4, 16, [262803244417256, 262803244417257, 262803244417258, 262803244417259, 262803244417260, 262803244417261, 262803244417262, 262803244417263, 262803244417264, 262803244417265, 262803244417266, 262803244417267, 262803244417268, 262803244417269, 396857386627072, 11112006825558016]),
(14, 5, 3, [329699565906945, 396857386627072, 11112006825558016]),
(14, 6, 16, [396857386627050, 396857386627051, 396857386627052, 396857386627053, 396857386627054, 396857386627055, 396857386627056, 396857386627057, 396857386627058, 396857386627059, 396857386627060, 396857386627061, 396857386627062, 396857386627063, 396857386627072, 11112006825558016]),
(14, 7, 1, [11112006825558016]),
(14, 8, 3, [10715149438930944, 10782307259651068, 11112006825558016]),
(14, 9, 16, [10715149438930944, 10849203581140717, 10849203581140718, 10849203581140719, 10849203581140720, 10849203581140721, 10849203581140722, 10849203581140723, 10849203581140724, 10849203581140725, 10849203581140726, 10849203581140727, 10849203581140728, 10849203581140729, 10849203581140730, 11112006825558016]),
(14, 10, 7, [10695883012804864, 10699756047226002, 10715149438930944, 10916407783803026, 11092740399431936, 11096613433853074, 11112006825558016]),
(14, 11, 37, [10695883012804864, 10705025526006767, 10705025526006768, 10705025526006769, 10705025526006770, 10705025526006771, 10705025526006772, 10705025526006773, 10705025526006774, 10705025526006775, 10705025526006776, 10705025526006777, 10705025526006778, 10705025526006779, 10705025526006780, 10714106056068352, 10714346330315919, 10715149438930944, 10983238748919806, 11092740399431936, 11101882912633839, 11101882912633840, 11101882912633841, 11101882912633842, 11101882912633843, 11101882912633844, 11101882912633845, 11101882912633846, 11101882912633847, 11101882912633848, 11101882912633849, 11101882912633850, 11101882912633851, 11101882912633852, 11110963442695424, 11111203716942991, 11112006825558016]),
(14, 12, 115, [10695883012804864, 10710296833687793, 10710296833687794, 10710296833687795, 10710296833687796, 10710296833687797, 10710296833687798, 10710296833687799, 10710296833687800, 10710296833687801, 10710296833687802, 10710296833687803, 10710296833687804, 10710296833687805, 10714106056068352, 10714764679641329, 10714764679641330, 10714764679641331, 10714764679641332, 10714764679641333, 10714764679641334, 10714764679641335, 10714764679641336, 10714764679641337, 10714764679641338, 10714764679641339, 10714764679641340, 10714764679641341, 10715090829022720, 10715118447360753, 10715118447360754, 10715118447360755, 10715118447360756, 10715118447360757, 10715118447360758, 10715118447360759, 10715118447360760, 10715118447360761, 10715118447360762, 10715118447360763, 10715118447360764, 10715118447360765, 10715145854871808, 10715146909006833, 10715146909006834, 10715146909006835, 10715146909006836, 10715146909006837, 10715146909006838, 10715146909006839, 10715146909006840, 10715146909006841, 10715146909006842, 10715146909006843, 10715146909006844, 10715146909006845, 10715149438930944, 11048572773736958, 11092740399431936, 11107154220314865, 11107154220314866, 11107154220314867, 11107154220314868, 11107154220314869, 11107154220314870, 11107154220314871, 11107154220314872, 11107154220314873, 11107154220314874, 11107154220314875, 11107154220314876, 11107154220314877, 11110963442695424, 11111622066268401, 11111622066268402, 11111622066268403, 11111622066268404, 11111622066268405, 11111622066268406, 11111622066268407, 11111622066268408, 11111622066268409, 11111622066268410, 11111622066268411, 11111622066268412, 11111622066268413, 11111948215649792, 11111975833987825, 11111975833987826, 11111975833987827, 11111975833987828, 11111975833987829, 11111975833987830, 11111975833987831, 11111975833987832, 11111975833987833, 11111975833987834, 11111975833987835, 11111975833987836, 11111975833987837, 11112003241498880, 11112004295633905, 11112004295633906, 11112004295633907, 11112004295633908, 11112004295633909, 11112004295633910, 11112004295633911, 11112004295633912, 11112004295633913, 11112004295633914, 11112004295633915, 11112004295633916, 11112004295633917, 11112006825558016]),
(14, 13, 1, [11112006825558016]),
]

"""#### 15-19"""

table_all_equal_by_base += [
(15, 0, 0, []),
(15, 1, 412, [1, 4754446846, 4754446847, 4754446848, 4754446849, 4754446850, 4754446851, 4754446852, 4754446853, 4754446854, 4754446855, 4754446856, 4754446857, 4754446858, 4754446859, 4754446860, 5771667729, 62220149971, 62220149972, 62220149973, 62220149974, 62220149975, 62220149976, 62220149977, 62220149978, 62220149979, 62220149980, 62220149981, 62220149982, 62220149983, 62220149984, 62220149985, 99003933354, 831087337471, 831087337472, 831087337473, 831087337474, 831087337475, 831087337476, 831087337477, 831087337478, 831087337479, 831087337480, 831087337481, 831087337482, 831087337483, 831087337484, 831087337485, 1729951171875, 1729951171876, 1734705618721, 1734705618722, 1734705618723, 1734705618724, 1734705618725, 1734705618726, 1734705618727, 1734705618728, 1734705618729, 1734705618730, 1734705618731, 1734705618732, 1734705618733, 1734705618734, 1734705618735, 1735722839604, 1792171321846, 1792171321847, 1792171321848, 1792171321849, 1792171321850, 1792171321851, 1792171321852, 1792171321853, 1792171321854, 1792171321855, 1792171321856, 1792171321857, 1792171321858, 1792171321859, 1792171321860, 1828955105229, 11210794368721, 11210794368722, 11210794368723, 11210794368724, 11210794368725, 11210794368726, 11210794368727, 11210794368728, 11210794368729, 11210794368730, 11210794368731, 11210794368732, 11210794368733, 11210794368734, 11210794368735, 32869072265625, 32869072265626, 32873826712471, 32873826712472, 32873826712473, 32873826712474, 32873826712475, 32873826712476, 32873826712477, 32873826712478, 32873826712479, 32873826712480, 32873826712481, 32873826712482, 32873826712483, 32873826712484, 32873826712485, 32874843933354, 32931292415596, 32931292415597, 32931292415598, 32931292415599, 32931292415600, 32931292415601, 32931292415602, 32931292415603, 32931292415604, 32931292415605, 32931292415606, 32931292415607, 32931292415608, 32931292415609, 32931292415610, 32968076198979, 153066790462471, 153066790462472, 153066790462473, 153066790462474, 153066790462475, 153066790462476, 153066790462477, 153066790462478, 153066790462479, 153066790462480, 153066790462481, 153066790462482, 153066790462483, 153066790462484, 153066790462485, 648731689453125, 648731689453126, 648736443899971, 648736443899972, 648736443899973, 648736443899974, 648736443899975, 648736443899976, 648736443899977, 648736443899978, 648736443899979, 648736443899980, 648736443899981, 648736443899982, 648736443899983, 648736443899984, 648736443899985, 648737461120854, 648793909603096, 648793909603097, 648793909603098, 648793909603099, 648793909603100, 648793909603101, 648793909603102, 648793909603103, 648793909603104, 648793909603105, 648793909603106, 648793909603107, 648793909603108, 648793909603109, 648793909603110, 648830693386479, 649562776790596, 649562776790597, 649562776790598, 649562776790599, 649562776790600, 649562776790601, 649562776790602, 649562776790603, 649562776790604, 649562776790605, 649562776790606, 649562776790607, 649562776790608, 649562776790609, 649562776790610, 650461640625000, 650461640625001, 650466395071846, 650466395071847, 650466395071848, 650466395071849, 650466395071850, 650466395071851, 650466395071852, 650466395071853, 650466395071854, 650466395071855, 650466395071856, 650466395071857, 650466395071858, 650466395071859, 650466395071860, 650467412292729, 650523860774971, 650523860774972, 650523860774973, 650523860774974, 650523860774975, 650523860774976, 650523860774977, 650523860774978, 650523860774979, 650523860774980, 650523860774981, 650523860774982, 650523860774983, 650523860774984, 650523860774985, 650560644558354, 659942483821846, 659942483821847, 659942483821848, 659942483821849, 659942483821850, 659942483821851, 659942483821852, 659942483821853, 659942483821854, 659942483821855, 659942483821856, 659942483821857, 659942483821858, 659942483821859, 659942483821860, 681600761718750, 681600761718751, 681605516165596, 681605516165597, 681605516165598, 681605516165599, 681605516165600, 681605516165601, 681605516165602, 681605516165603, 681605516165604, 681605516165605, 681605516165606, 681605516165607, 681605516165608, 681605516165609, 681605516165610, 681606533386479, 681662981868721, 681662981868722, 681662981868723, 681662981868724, 681662981868725, 681662981868726, 681662981868727, 681662981868728, 681662981868729, 681662981868730, 681662981868731, 681662981868732, 681662981868733, 681662981868734, 681662981868735, 681699765652104, 2160709013671873, 14920828857421875, 14920828857421876, 14920833611868721, 14920833611868722, 14920833611868723, 14920833611868724, 14920833611868725, 14920833611868726, 14920833611868727, 14920833611868728, 14920833611868729, 14920833611868730, 14920833611868731, 14920833611868732, 14920833611868733, 14920833611868734, 14920833611868735, 14920834629089604, 14920891077571846, 14920891077571847, 14920891077571848, 14920891077571849, 14920891077571850, 14920891077571851, 14920891077571852, 14920891077571853, 14920891077571854, 14920891077571855, 14920891077571856, 14920891077571857, 14920891077571858, 14920891077571859, 14920891077571860, 14920927861355229, 14921659944759346, 14921659944759347, 14921659944759348, 14921659944759349, 14921659944759350, 14921659944759351, 14921659944759352, 14921659944759353, 14921659944759354, 14921659944759355, 14921659944759356, 14921659944759357, 14921659944759358, 14921659944759359, 14921659944759360, 14922558808593750, 14922558808593751, 14922563563040596, 14922563563040597, 14922563563040598, 14922563563040599, 14922563563040600, 14922563563040601, 14922563563040602, 14922563563040603, 14922563563040604, 14922563563040605, 14922563563040606, 14922563563040607, 14922563563040608, 14922563563040609, 14922563563040610, 14922564580261479, 14922621028743721, 14922621028743722, 14922621028743723, 14922621028743724, 14922621028743725, 14922621028743726, 14922621028743727, 14922621028743728, 14922621028743729, 14922621028743730, 14922621028743731, 14922621028743732, 14922621028743733, 14922621028743734, 14922621028743735, 14922657812527104, 14932039651790596, 14932039651790597, 14932039651790598, 14932039651790599, 14932039651790600, 14932039651790601, 14932039651790602, 14932039651790603, 14932039651790604, 14932039651790605, 14932039651790606, 14932039651790607, 14932039651790608, 14932039651790609, 14932039651790610, 14953697929687500, 14953697929687501, 14953702684134346, 14953702684134347, 14953702684134348, 14953702684134349, 14953702684134350, 14953702684134351, 14953702684134352, 14953702684134353, 14953702684134354, 14953702684134355, 14953702684134356, 14953702684134357, 14953702684134358, 14953702684134359, 14953702684134360, 14953703701355229, 14953760149837471, 14953760149837472, 14953760149837473, 14953760149837474, 14953760149837475, 14953760149837476, 14953760149837477, 14953760149837478, 14953760149837479, 14953760149837480, 14953760149837481, 14953760149837482, 14953760149837483, 14953760149837484, 14953760149837485, 14953796933620854, 31278135027204240]),
(15, 2, 73, [1729951171847, 1729951171848, 1729951171849, 1729951171850, 1729951171851, 1729951171852, 1729951171853, 1729951171854, 1729951171855, 1729951171856, 1729951171857, 1729951171858, 1729951171859, 1729951171860, 1729951171861, 1729951171875, 23320457687768, 32869072265625, 318311015624972, 318311015624973, 318311015624974, 318311015624975, 318311015624976, 318311015624977, 318311015624978, 318311015624979, 318311015624980, 318311015624981, 318311015624982, 318311015624983, 318311015624984, 318311015624985, 318311015624986, 648731689453125, 650461640624972, 650461640624973, 650461640624974, 650461640624975, 650461640624976, 650461640624977, 650461640624978, 650461640624979, 650461640624980, 650461640624981, 650461640624982, 650461640624983, 650461640624984, 650461640624985, 650461640624986, 650461640625000, 672052147140893, 681600761718750, 4375877606125268, 14920828857421875, 14922558808593722, 14922558808593723, 14922558808593724, 14922558808593725, 14922558808593726, 14922558808593727, 14922558808593728, 14922558808593729, 14922558808593730, 14922558808593731, 14922558808593732, 14922558808593733, 14922558808593734, 14922558808593735, 14922558808593736, 14922558808593750, 14944149315109643, 14953697929687500, 437893890380859375]),
(15, 3, 4, [648731689453125, 6646808368828123, 14920828857421875, 437893890380859375]),
(15, 4, 19, [648731689453099, 648731689453100, 648731689453101, 648731689453102, 648731689453103, 648731689453104, 648731689453105, 648731689453106, 648731689453107, 648731689453108, 648731689453109, 648731689453110, 648731689453111, 648731689453112, 648731689453113, 648731689453125, 8916999529696873, 14920828857421875, 437893890380859375]),
(15, 5, 2, [14920828857421875, 437893890380859375]),
(15, 6, 5, [13458130908046873, 14272097167968750, 14437283271328123, 14920828857421875, 437893890380859375]),
(15, 7, 1, [437893890380859375]),
(15, 8, 5, [422973061523437500, 423456607109531248, 423621793212890625, 424435759472812498, 437893890380859375]),
(15, 9, 19, [422973061523437500, 423621793212890604, 423621793212890605, 423621793212890606, 423621793212890607, 423621793212890608, 423621793212890609, 423621793212890610, 423621793212890611, 423621793212890612, 423621793212890613, 423621793212890614, 423621793212890615, 423621793212890616, 423621793212890617, 423621793212890618, 423621793212890625, 426700274995068748, 437893890380859375]),
(15, 10, 3, [422973061523437500, 428976890851162498, 437893890380859375]),
(15, 11, 4, [422973061523437500, 431247082012031248, 437245158691406250, 437893890380859375]),
(15, 12, 25, [422940192451171875, 422949741065749728, 422973061523437500, 433518012774734103, 437212289619140625, 437221838233718478, 437245158691406250, 437575579365234357, 437575579365234358, 437575579365234359, 437575579365234360, 437575579365234361, 437575579365234362, 437575579365234363, 437575579365234364, 437575579365234365, 437575579365234366, 437575579365234367, 437575579365234368, 437575579365234369, 437575579365234370, 437575579365234371, 437861021308593750, 437870569923171603, 437893890380859375]),
(15, 13, 378, [422940093447238521, 422940130231021859, 422940130231021860, 422940130231021861, 422940130231021862, 422940130231021863, 422940130231021864, 422940130231021865, 422940130231021866, 422940130231021867, 422940130231021868, 422940130231021869, 422940130231021870, 422940130231021871, 422940130231021872, 422940186679504146, 422940187696724984, 422940187696724985, 422940187696724986, 422940187696724987, 422940187696724988, 422940187696724989, 422940187696724990, 422940187696724991, 422940187696724992, 422940187696724993, 422940187696724994, 422940187696724995, 422940187696724996, 422940187696724997, 422940192451171875, 422961850729068734, 422961850729068735, 422961850729068736, 422961850729068737, 422961850729068738, 422961850729068739, 422961850729068740, 422961850729068741, 422961850729068742, 422961850729068743, 422961850729068744, 422961850729068745, 422961850729068746, 422961850729068747, 422971232568332271, 422971269352115609, 422971269352115610, 422971269352115611, 422971269352115612, 422971269352115613, 422971269352115614, 422971269352115615, 422971269352115616, 422971269352115617, 422971269352115618, 422971269352115619, 422971269352115620, 422971269352115621, 422971269352115622, 422971325800597896, 422971326817818734, 422971326817818735, 422971326817818736, 422971326817818737, 422971326817818738, 422971326817818739, 422971326817818740, 422971326817818741, 422971326817818742, 422971326817818743, 422971326817818744, 422971326817818745, 422971326817818746, 422971326817818747, 422971331572265625, 422972230436099984, 422972230436099985, 422972230436099986, 422972230436099987, 422972230436099988, 422972230436099989, 422972230436099990, 422972230436099991, 422972230436099992, 422972230436099993, 422972230436099994, 422972230436099995, 422972230436099996, 422972230436099997, 422972962519504146, 422972999303287484, 422972999303287485, 422972999303287486, 422972999303287487, 422972999303287488, 422972999303287489, 422972999303287490, 422972999303287491, 422972999303287492, 422972999303287493, 422972999303287494, 422972999303287495, 422972999303287496, 422972999303287497, 422973055751769771, 422973056768990609, 422973056768990610, 422973056768990611, 422973056768990612, 422973056768990613, 422973056768990614, 422973056768990615, 422973056768990616, 422973056768990617, 422973056768990618, 422973056768990619, 422973056768990620, 422973056768990621, 422973056768990622, 422973061523437500, 435733181367187498, 437212190615207271, 437212227398990609, 437212227398990610, 437212227398990611, 437212227398990612, 437212227398990613, 437212227398990614, 437212227398990615, 437212227398990616, 437212227398990617, 437212227398990618, 437212227398990619, 437212227398990620, 437212227398990621, 437212227398990622, 437212283847472896, 437212284864693734, 437212284864693735, 437212284864693736, 437212284864693737, 437212284864693738, 437212284864693739, 437212284864693740, 437212284864693741, 437212284864693742, 437212284864693743, 437212284864693744, 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39305924754756031254525, 39305924761103528546304, 39305926553745208318846, 39305928062595557431296, 39305928128847560414189, 39305928128847560414190, 39305928128847560414191, 39305928128847560414192, 39305928128847560414193, 39305928128847560414194, 39305928128847560414195, 39305928128847560414196, 39305928128847560414197, 39305928128847560414198, 39305928128847560414199, 39305928128847560414200, 39305928128847560414201, 39305928128847560414202, 39305928128847560414203, 39305928128847560414204, 39305928128847560414205, 39305928223266582629376, 39305928225250175533037, 39305928225250175533038, 39305928225250175533039, 39305928225250175533040, 39305928225250175533041, 39305928225250175533042, 39305928225250175533043, 39305928225250175533044, 39305928225250175533045, 39305928225250175533046, 39305928225250175533047, 39305928225250175533048, 39305928225250175533049, 39305928225250175533050, 39305928225250175533051, 39305928225250175533052, 39305928225250175533053, 39305928231597672824832, 39338710519284527800301, 39338710519284527800302, 39338710519284527800303, 39338710519284527800304, 39338710519284527800305, 39338710519284527800306, 39338710519284527800307, 39338710519284527800308, 39338710519284527800309, 39338710519284527800310, 39338710519284527800311, 39338710519284527800312, 39338710519284527800313, 39338710519284527800314, 39338710519284527800315, 39338710519284527800316, 39338710519284527800317, 39344683070704715753472, 39344683136956718736365, 39344683136956718736366, 39344683136956718736367, 39344683136956718736368, 39344683136956718736369, 39344683136956718736370, 39344683136956718736371, 39344683136956718736372, 39344683136956718736373, 39344683136956718736374, 39344683136956718736375, 39344683136956718736376, 39344683136956718736377, 39344683136956718736378, 39344683136956718736379, 39344683136956718736380, 39344683136956718736381, 39344683231375740951552, 39344683233359333855213, 39344683233359333855214, 39344683233359333855215, 39344683233359333855216, 39344683233359333855217, 39344683233359333855218, 39344683233359333855219, 39344683233359333855220, 39344683233359333855221, 39344683233359333855222, 39344683233359333855223, 39344683233359333855224, 39344683233359333855225, 39344683233359333855226, 39344683233359333855227, 39344683233359333855228, 39344683233359333855229, 39344683239706831147008, 39345947897482040471422, 39346331555423248054272, 39346331621675251037165, 39346331621675251037166, 39346331621675251037167, 39346331621675251037168, 39346331621675251037169, 39346331621675251037170, 39346331621675251037171, 39346331621675251037172, 39346331621675251037173, 39346331621675251037174, 39346331621675251037175, 39346331621675251037176, 39346331621675251037177, 39346331621675251037178, 39346331621675251037179, 39346331621675251037180, 39346331621675251037181, 39346331716094273252352, 39346331718077866156013, 39346331718077866156014, 39346331718077866156015, 39346331718077866156016, 39346331718077866156017, 39346331718077866156018, 39346331718077866156019, 39346331718077866156020, 39346331718077866156021, 39346331718077866156022, 39346331718077866156023, 39346331718077866156024, 39346331718077866156025, 39346331718077866156026, 39346331718077866156027, 39346331718077866156028, 39346331718077866156029, 39346331724425363447808, 39346380311343383347181, 39346380311343383347182, 39346380311343383347183, 39346380311343383347184, 39346380311343383347185, 39346380311343383347186, 39346380311343383347187, 39346380311343383347188, 39346380311343383347189, 39346380311343383347190, 39346380311343383347191, 39346380311343383347192, 39346380311343383347193, 39346380311343383347194, 39346380311343383347195, 39346380311343383347196, 39346380311343383347197, 39346404435800277903360, 39346404502052280886253, 39346404502052280886254, 39346404502052280886255, 39346404502052280886256, 39346404502052280886257, 39346404502052280886258, 39346404502052280886259, 39346404502052280886260, 39346404502052280886261, 39346404502052280886262, 39346404502052280886263, 39346404502052280886264, 39346404502052280886265, 39346404502052280886266, 39346404502052280886267, 39346404502052280886268, 39346404502052280886269, 39346404596471303101440, 39346404598454896005101, 39346404598454896005102, 39346404598454896005103, 39346404598454896005104, 39346404598454896005105, 39346404598454896005106, 39346404598454896005107, 39346404598454896005108, 39346404598454896005109, 39346404598454896005110, 39346404598454896005111, 39346404598454896005112, 39346404598454896005113, 39346404598454896005114, 39346404598454896005115, 39346404598454896005116, 39346404598454896005117, 39346404604802393296896, 39346406397444073069438, 39346407906294422181888, 39346407972546425164781, 39346407972546425164782, 39346407972546425164783, 39346407972546425164784, 39346407972546425164785, 39346407972546425164786, 39346407972546425164787, 39346407972546425164788, 39346407972546425164789, 39346407972546425164790, 39346407972546425164791, 39346407972546425164792, 39346407972546425164793, 39346407972546425164794, 39346407972546425164795, 39346407972546425164796, 39346407972546425164797, 39346408066965447379968, 39346408068949040283629, 39346408068949040283630, 39346408068949040283631, 39346408068949040283632, 39346408068949040283633, 39346408068949040283634, 39346408068949040283635, 39346408068949040283636, 39346408068949040283637, 39346408068949040283638, 39346408068949040283639, 39346408068949040283640, 39346408068949040283641, 39346408068949040283642, 39346408068949040283643, 39346408068949040283644, 39346408068949040283645, 39346408075296537575424]),
(18, 17, 1, [39346408075296537575424]),
(19, 0, 0, []),
(19, 1, 15, [1, 13494919482970, 197523599065350, 285110156562925, 3412946357863637, 6096745366999749, 59549673273552806, 138338944756239663, 1044276562032326420, 3064069652457272800, 72980189551829521262, 326362533851172735512, 1876470249947950971244, 53003442302023974393769, 109912203092239643840220]),
(19, 2, 13, [122536147345585331, 138338944756076651, 2150084790496538592, 3064069652453974022, 37921120283586881264, 72980189551829513200, 672269454098701945236, 1876470249947950971244, 11979731054123624988741, 53003442301994268308464, 53003442302003222893412, 53003442302023974393769, 1978419655660313589123979]),
(19, 3, 7, [57465296623637492013, 72979917936525089557, 72980189551829513200, 1876470249947950971244, 18152262531480006035891, 53003442301994268308063, 1978419655660313589123979]),
(19, 4, 7, [1365193971372874995805, 1876470249947950971244, 53000378517451970888862, 53000378517451970888863, 53000570042584482426493, 53003442301994268308063, 1978419655660313589123979]),
(19, 5, 14, [1711104305065560619656, 1803490332012803749475, 1803490340984773180765, 1803490345506278183960, 1803628399342252095643, 1807602709530190678530, 1876470249947950970843, 30478292183823772865916, 53000378245836663158400, 53000378517450525840799, 53000378517451970888862, 53001616036860183772290, 53003442301994200956714, 1978419655660313589123979]),
(19, 6, 7, [36651543627512253609944, 53000378245836663158400, 53000378492995793798669, 53000378517450525840799, 53002721787806379184601, 53003442301994200956714, 1978419655660313589123979]),
(19, 7, 13, [42808006784347146759384, 51126972052076023422525, 51199813916146868343619, 51199813916176571130547, 51199819244450044596124, 51199819714316861921832, 51199819721986569715520, 51199819727810269167558, 51199819727810269167559, 51199951969981397370230, 51380492945149645429837, 53000378245836663158400, 1978419655660313589123979]),
(19, 8, 13, [51126972052076023422525, 51188608594699226710084, 51196888185439096652272, 51197275090492835805979, 51199813631066417874483, 51199813804749798729106, 51199813916146797693492, 51199951969981397370230, 51727507117382228422909, 53000378245835221580168, 53000378245835439194971, 53000378245836663158400, 1978419655660313589123979]),
(19, 9, 1, [1978419655660313589123979]),
(19, 10, 12, [1925419277414476925965579, 1925419277414478149929004, 1925419277414478367543811, 1927219703690332191753749, 1927219841744166791430487, 1927219841855563790394869, 1927219842029247171249496, 1927222380569820753317996, 1927222767474874492471707, 1927231047065614362413892, 1927292683608237565701454, 1978419655660313589123979]),
(19, 11, 12, [1925419277414476925965579, 1927039162715163943694138, 1927219703690332191753749, 1927219835932503319956421, 1927219835938327019408455, 1927219835945996727202147, 1927219836415863544527851, 1927219841744137017993432, 1927219841744166720780360, 1927250591232592802169502, 1927292683608237565701454, 1978419655660313589123979]),
(19, 12, 7, [1925416213358319388167265, 1925416933872507209939374, 1925419277142863063283180, 1925419277167317795325306, 1925419277414476925965579, 1941768112032801335514031, 1978419655660313589123979]),
(19, 13, 15, [1925416213358319388167265, 1925418039623453405351685, 1925419277142861618235117, 1925419277142863063283180, 1925419277414476925965579, 1947941363476489816258059, 1976543185410365638153136, 1976612052950783398445445, 1976616027260971337028336, 1976616052300624275784821, 1976616165314807310940019, 1976616165319328815943211, 1976616165328300785374504, 1976708551355248028504319, 1978419655660313589123979]),
(19, 14, 8, [1925416213358319320815916, 1925416213358319320815917, 1925419085617729106697482, 1925419277142861618235117, 1954113627008920910391575, 1976543185410365638152735, 1977054461688940714128170, 1978419655660313589123979]),
(19, 15, 7, [1925416213358319320815916, 1960267393128833583088084, 1976543185410365638152735, 1978346675470761759610779, 1978346675742377064034422, 1978362190363689951631962, 1978419655660313589123979]),
(19, 16, 12, [1925416213358289614730210, 1925416213358310366230564, 1925416213358319320815515, 1966439924606189964135235, 1976543185410365638152735, 1977747386206214887178739, 1978346675470761759610779, 1978381734540030002242711, 1978416591590661135149957, 1978417505575523092585384, 1978419517321368833047328, 1978419655660313589123979]),
(19, 17, 17, [1925416213358289614730210, 1976543185410365638152735, 1978346675470761759602717, 1978401229680167227357438, 1978416591590661131851179, 1978418611383751556797555, 1978419517321368832884316, 1978419596110640315571169, 1978419649563568222124230, 1978419652247367231260338, 1978419655375203432561054, 1978419655375203432561055, 1978419655375203432561074, 1978419655375203432561075, 1978419655462789990058625, 1978419655646818669641009, 1978419655660313589123979]),
(19, 18, 1, [1978419655660313589123979]),
]

"""#### 20"""

table_all_equal_by_base += [
(20, 0, 0, []),
(20, 1, 1545, [1, 20479999999961, 20479999999962, 20479999999963, 20479999999964, 20479999999965, 20479999999966, 20479999999967, 20479999999968, 20479999999969, 20479999999970, 20479999999971, 20479999999972, 20479999999973, 20479999999974, 20479999999975, 20479999999976, 20479999999977, 20479999999978, 20479999999979, 20479999999980, 20480000000000, 20480000000001, 368639999999961, 368639999999962, 368639999999963, 368639999999964, 368639999999965, 368639999999966, 368639999999967, 368639999999968, 368639999999969, 368639999999970, 368639999999971, 368639999999972, 368639999999973, 368639999999974, 368639999999975, 368639999999976, 368639999999977, 368639999999978, 368639999999979, 368639999999980, 471040000000000, 471040000000001, 6696959999999961, 6696959999999962, 6696959999999963, 6696959999999964, 6696959999999965, 6696959999999966, 6696959999999967, 6696959999999968, 6696959999999969, 6696959999999970, 6696959999999971, 6696959999999972, 6696959999999973, 6696959999999974, 6696959999999975, 6696959999999976, 6696959999999977, 6696959999999978, 6696959999999979, 6696959999999980, 10485760000000000, 10485760000000001, 122879999999999961, 122879999999999962, 122879999999999963, 122879999999999964, 122879999999999965, 122879999999999966, 122879999999999967, 122879999999999968, 122879999999999969, 122879999999999970, 122879999999999971, 122879999999999972, 122879999999999973, 122879999999999974, 122879999999999975, 122879999999999976, 122879999999999977, 122879999999999978, 122879999999999979, 122879999999999980, 235765760000000000, 235765760000000001, 2266193919999999961, 2266193919999999962, 2266193919999999963, 2266193919999999964, 2266193919999999965, 2266193919999999966, 2266193919999999967, 2266193919999999968, 2266193919999999969, 2266193919999999970, 2266193919999999971, 2266193919999999972, 2266193919999999973, 2266193919999999974, 2266193919999999975, 2266193919999999976, 2266193919999999977, 2266193919999999978, 2266193919999999979, 2266193919999999980, 5622005760000000000, 5622005760000000001, 42093096959999999961, 42093096959999999962, 42093096959999999963, 42093096959999999964, 42093096959999999965, 42093096959999999966, 42093096959999999967, 42093096959999999968, 42093096959999999969, 42093096959999999970, 42093096959999999971, 42093096959999999972, 42093096959999999973, 42093096959999999974, 42093096959999999975, 42093096959999999976, 42093096959999999977, 42093096959999999978, 42093096959999999979, 42093096959999999980, 131072000000000000000, 131072000000000000001, 131072020479999999961, 131072020479999999962, 131072020479999999963, 131072020479999999964, 131072020479999999965, 131072020479999999966, 131072020479999999967, 131072020479999999968, 131072020479999999969, 131072020479999999970, 131072020479999999971, 131072020479999999972, 131072020479999999973, 131072020479999999974, 131072020479999999975, 131072020479999999976, 131072020479999999977, 131072020479999999978, 131072020479999999979, 131072020479999999980, 131072020480000000000, 131072020480000000001, 131072368639999999961, 131072368639999999962, 131072368639999999963, 131072368639999999964, 131072368639999999965, 131072368639999999966, 131072368639999999967, 131072368639999999968, 131072368639999999969, 131072368639999999970, 131072368639999999971, 131072368639999999972, 131072368639999999973, 131072368639999999974, 131072368639999999975, 131072368639999999976, 131072368639999999977, 131072368639999999978, 131072368639999999979, 131072368639999999980, 131072471040000000000, 131072471040000000001, 131078696959999999961, 131078696959999999962, 131078696959999999963, 131078696959999999964, 131078696959999999965, 131078696959999999966, 131078696959999999967, 131078696959999999968, 131078696959999999969, 131078696959999999970, 131078696959999999971, 131078696959999999972, 131078696959999999973, 131078696959999999974, 131078696959999999975, 131078696959999999976, 131078696959999999977, 131078696959999999978, 131078696959999999979, 131078696959999999980, 131082485760000000000, 131082485760000000001, 131194879999999999961, 131194879999999999962, 131194879999999999963, 131194879999999999964, 131194879999999999965, 131194879999999999966, 131194879999999999967, 131194879999999999968, 131194879999999999969, 131194879999999999970, 131194879999999999971, 131194879999999999972, 131194879999999999973, 131194879999999999974, 131194879999999999975, 131194879999999999976, 131194879999999999977, 131194879999999999978, 131194879999999999979, 131194879999999999980, 131307765760000000000, 131307765760000000001, 133338193919999999961, 133338193919999999962, 133338193919999999963, 133338193919999999964, 133338193919999999965, 133338193919999999966, 133338193919999999967, 133338193919999999968, 133338193919999999969, 133338193919999999970, 133338193919999999971, 133338193919999999972, 133338193919999999973, 133338193919999999974, 133338193919999999975, 133338193919999999976, 133338193919999999977, 133338193919999999978, 133338193919999999979, 133338193919999999980, 136694005760000000000, 136694005760000000001, 786431999999999999961, 786431999999999999962, 786431999999999999963, 786431999999999999964, 786431999999999999965, 786431999999999999966, 786431999999999999967, 786431999999999999968, 786431999999999999969, 786431999999999999970, 786431999999999999971, 786431999999999999972, 786431999999999999973, 786431999999999999974, 786431999999999999975, 786431999999999999976, 786431999999999999977, 786431999999999999978, 786431999999999999979, 786431999999999999980, 3276800000000000000000, 3276800000000000000001, 3276800020479999999961, 3276800020479999999962, 3276800020479999999963, 3276800020479999999964, 3276800020479999999965, 3276800020479999999966, 3276800020479999999967, 3276800020479999999968, 3276800020479999999969, 3276800020479999999970, 3276800020479999999971, 3276800020479999999972, 3276800020479999999973, 3276800020479999999974, 3276800020479999999975, 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104854192127631359999999985, 104854192127631359999999986, 104854192127631359999999987, 104854192127631359999999988, 104854192127631359999999989, 104854192127631359999999990, 104854192127631359999999991, 104854192127631359999999992, 104854192127631359999999993, 104854192127631359999999994, 104854192127631359999999995, 104854192127631359999999996, 104854192127631359999999997, 104854192128000000000000000, 104854281106903039999999979, 104854281106903039999999980, 104854281106903039999999981, 104854281106903039999999982, 104854281106903039999999983, 104854281106903039999999984, 104854281106903039999999985, 104854281106903039999999986, 104854281106903039999999987, 104854281106903039999999988, 104854281106903039999999989, 104854281106903039999999990, 104854281106903039999999991, 104854281106903039999999992, 104854281106903039999999993, 104854281106903039999999994, 104854281106903039999999995, 104854281106903039999999996, 104854281106903039999999997, 104854317577994240000000000, 104854320933806079999999979, 104854320933806079999999980, 104854320933806079999999981, 104854320933806079999999982, 104854320933806079999999983, 104854320933806079999999984, 104854320933806079999999985, 104854320933806079999999986, 104854320933806079999999987, 104854320933806079999999988, 104854320933806079999999989, 104854320933806079999999990, 104854320933806079999999991, 104854320933806079999999992, 104854320933806079999999993, 104854320933806079999999994, 104854320933806079999999995, 104854320933806079999999996, 104854320933806079999999997, 104854322964234240000000000, 104854323077119999999999979, 104854323077119999999999980, 104854323077119999999999981, 104854323077119999999999982, 104854323077119999999999983, 104854323077119999999999984, 104854323077119999999999985, 104854323077119999999999986, 104854323077119999999999987, 104854323077119999999999988, 104854323077119999999999989, 104854323077119999999999990, 104854323077119999999999991, 104854323077119999999999992, 104854323077119999999999993, 104854323077119999999999994, 104854323077119999999999995, 104854323077119999999999996, 104854323077119999999999997, 104854323189514240000000000, 104854323193303039999999979, 104854323193303039999999980, 104854323193303039999999981, 104854323193303039999999982, 104854323193303039999999983, 104854323193303039999999984, 104854323193303039999999985, 104854323193303039999999986, 104854323193303039999999987, 104854323193303039999999988, 104854323193303039999999989, 104854323193303039999999990, 104854323193303039999999991, 104854323193303039999999992, 104854323193303039999999993, 104854323193303039999999994, 104854323193303039999999995, 104854323193303039999999996, 104854323193303039999999997, 104854323199528960000000000, 104854323199631359999999979, 104854323199631359999999980, 104854323199631359999999981, 104854323199631359999999982, 104854323199631359999999983, 104854323199631359999999984, 104854323199631359999999985, 104854323199631359999999986, 104854323199631359999999987, 104854323199631359999999988, 104854323199631359999999989, 104854323199631359999999990, 104854323199631359999999991, 104854323199631359999999992, 104854323199631359999999993, 104854323199631359999999994, 104854323199631359999999995, 104854323199631359999999996, 104854323199631359999999997, 104854323200000000000000000, 104856813567999999999999979, 104856813567999999999999980, 104856813567999999999999981, 104856813567999999999999982, 104856813567999999999999983, 104856813567999999999999984, 104856813567999999999999985, 104856813567999999999999986, 104856813567999999999999987, 104856813567999999999999988, 104856813567999999999999989, 104856813567999999999999990, 104856813567999999999999991, 104856813567999999999999992, 104856813567999999999999993, 104856813567999999999999994, 104856813567999999999999995, 104856813567999999999999996, 104856813567999999999999997, 104857463305994240000000000, 104857466661806079999999979, 104857466661806079999999980, 104857466661806079999999981, 104857466661806079999999982, 104857466661806079999999983, 104857466661806079999999984, 104857466661806079999999985, 104857466661806079999999986, 104857466661806079999999987, 104857466661806079999999988, 104857466661806079999999989, 104857466661806079999999990, 104857466661806079999999991, 104857466661806079999999992, 104857466661806079999999993, 104857466661806079999999994, 104857466661806079999999995, 104857466661806079999999996, 104857466661806079999999997, 104857468692234240000000000, 104857468805119999999999979, 104857468805119999999999980, 104857468805119999999999981, 104857468805119999999999982, 104857468805119999999999983, 104857468805119999999999984, 104857468805119999999999985, 104857468805119999999999986, 104857468805119999999999987, 104857468805119999999999988, 104857468805119999999999989, 104857468805119999999999990, 104857468805119999999999991, 104857468805119999999999992, 104857468805119999999999993, 104857468805119999999999994, 104857468805119999999999995, 104857468805119999999999996, 104857468805119999999999997, 104857468917514240000000000, 104857468921303039999999979, 104857468921303039999999980, 104857468921303039999999981, 104857468921303039999999982, 104857468921303039999999983, 104857468921303039999999984, 104857468921303039999999985, 104857468921303039999999986, 104857468921303039999999987, 104857468921303039999999988, 104857468921303039999999989, 104857468921303039999999990, 104857468921303039999999991, 104857468921303039999999992, 104857468921303039999999993, 104857468921303039999999994, 104857468921303039999999995, 104857468921303039999999996, 104857468921303039999999997, 104857468927528960000000000, 104857468927631359999999979, 104857468927631359999999980, 104857468927631359999999981, 104857468927631359999999982, 104857468927631359999999983, 104857468927631359999999984, 104857468927631359999999985, 104857468927631359999999986, 104857468927631359999999987, 104857468927631359999999988, 104857468927631359999999989, 104857468927631359999999990, 104857468927631359999999991, 104857468927631359999999992, 104857468927631359999999993, 104857468927631359999999994, 104857468927631359999999995, 104857468927631359999999996, 104857468927631359999999997, 104857468928000000000000000, 104857557906903039999999979, 104857557906903039999999980, 104857557906903039999999981, 104857557906903039999999982, 104857557906903039999999983, 104857557906903039999999984, 104857557906903039999999985, 104857557906903039999999986, 104857557906903039999999987, 104857557906903039999999988, 104857557906903039999999989, 104857557906903039999999990, 104857557906903039999999991, 104857557906903039999999992, 104857557906903039999999993, 104857557906903039999999994, 104857557906903039999999995, 104857557906903039999999996, 104857557906903039999999997, 104857594377994240000000000, 104857597733806079999999979, 104857597733806079999999980, 104857597733806079999999981, 104857597733806079999999982, 104857597733806079999999983, 104857597733806079999999984, 104857597733806079999999985, 104857597733806079999999986, 104857597733806079999999987, 104857597733806079999999988, 104857597733806079999999989, 104857597733806079999999990, 104857597733806079999999991, 104857597733806079999999992, 104857597733806079999999993, 104857597733806079999999994, 104857597733806079999999995, 104857597733806079999999996, 104857597733806079999999997, 104857599764234240000000000, 104857599877119999999999979, 104857599877119999999999980, 104857599877119999999999981, 104857599877119999999999982, 104857599877119999999999983, 104857599877119999999999984, 104857599877119999999999985, 104857599877119999999999986, 104857599877119999999999987, 104857599877119999999999988, 104857599877119999999999989, 104857599877119999999999990, 104857599877119999999999991, 104857599877119999999999992, 104857599877119999999999993, 104857599877119999999999994, 104857599877119999999999995, 104857599877119999999999996, 104857599877119999999999997, 104857599989514240000000000, 104857599993303039999999979, 104857599993303039999999980, 104857599993303039999999981, 104857599993303039999999982, 104857599993303039999999983, 104857599993303039999999984, 104857599993303039999999985, 104857599993303039999999986, 104857599993303039999999987, 104857599993303039999999988, 104857599993303039999999989, 104857599993303039999999990, 104857599993303039999999991, 104857599993303039999999992, 104857599993303039999999993, 104857599993303039999999994, 104857599993303039999999995, 104857599993303039999999996, 104857599993303039999999997, 104857599999528960000000000, 104857599999631359999999979, 104857599999631359999999980, 104857599999631359999999981, 104857599999631359999999982, 104857599999631359999999983, 104857599999631359999999984, 104857599999631359999999985, 104857599999631359999999986, 104857599999631359999999987, 104857599999631359999999988, 104857599999631359999999989, 104857599999631359999999990, 104857599999631359999999991, 104857599999631359999999992, 104857599999631359999999993, 104857599999631359999999994, 104857599999631359999999995, 104857599999631359999999996, 104857599999631359999999997, 104857600000000000000000000]),
(20, 19, 1, [104857600000000000000000000]),
]

"""#### 21"""

table_all_equal_by_base += [
(21, 0, 0, []),
(21, 1, 1219, [1, 764527028941797, 12678492621088309, 12678492621088310, 12678492621088311, 12678492621088312, 12678492621088313, 12678492621088314, 12678492621088315, 12678492621088316, 12678492621088317, 12678492621088318, 12678492621088319, 12678492621088320, 12678492621088321, 12678492621088322, 12678492621088323, 12678492621088324, 12678492621088325, 12678492621088326, 12678492621088327, 12678492621088328, 12678492621088329, 17673771229573524, 244072921471344715, 244072921471344716, 244072921471344717, 244072921471344718, 244072921471344719, 244072921471344720, 244072921471344721, 244072921471344722, 244072921471344723, 244072921471344724, 244072921471344725, 244072921471344726, 244072921471344727, 244072921471344728, 244072921471344729, 244072921471344730, 244072921471344731, 244072921471344732, 244072921471344733, 244072921471344734, 244072921471344735, 411926340637651896, 411926340637651897, 412690867666593693, 4737152917332996763, 4737152917332996764, 4737152917332996765, 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9975832719035870762, 9975832719035870763, 9975832719035870764, 9975832719035870765, 9975832719035870766, 9975832719035870767, 9975832719035870768, 9975832719035870769, 9975832719035870770, 9975832719035870771, 9975832719035870772, 9975832719035870773, 9975832719035870774, 9975832719035870775, 9975832719035870776, 9975832719035870777, 9975832719035870778, 10143686138202177939, 10143686138202177940, 10144450665231119736, 92322991095413731150, 92322991095413731151, 92322991095413731152, 92322991095413731153, 92322991095413731154, 92322991095413731155, 92322991095413731156, 92322991095413731157, 92322991095413731158, 92322991095413731159, 92322991095413731160, 92322991095413731161, 92322991095413731162, 92322991095413731163, 92322991095413731164, 92322991095413731165, 92322991095413731166, 92322991095413731167, 92322991095413731168, 92322991095413731169, 92322991095413731170, 243293994939113151075, 243293994939113151076, 243294759466142092872, 243306673431734239384, 243306673431734239385, 243306673431734239386, 243306673431734239387, 243306673431734239388, 243306673431734239389, 243306673431734239390, 243306673431734239391, 243306673431734239392, 243306673431734239393, 243306673431734239394, 243306673431734239395, 243306673431734239396, 243306673431734239397, 243306673431734239398, 243306673431734239399, 243306673431734239400, 243306673431734239401, 243306673431734239402, 243306673431734239403, 243306673431734239404, 243311668710342724599, 243538067860584495790, 243538067860584495791, 243538067860584495792, 243538067860584495793, 243538067860584495794, 243538067860584495795, 243538067860584495796, 243538067860584495797, 243538067860584495798, 243538067860584495799, 243538067860584495800, 243538067860584495801, 243538067860584495802, 243538067860584495803, 243538067860584495804, 243538067860584495805, 243538067860584495806, 243538067860584495807, 243538067860584495808, 243538067860584495809, 243538067860584495810, 243705921279750802971, 243705921279750802972, 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(21, 20, 1, [5842587018385982521381124421]),
]

"""#### 22"""

table_all_equal_by_base += [
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(22, 21, 1, [341427877364219557396646723584]),
]

"""#### 23 (pretty short)"""

table_all_equal_by_base += [
(23, 0, 0, []),
(23, 1, 21, [1, 41826814261329722, 47495558901969667, 880488189921130452, 1192553866711364267, 18692510479760064900, 30150934005899904944, 399489596502465285534, 770947400346194265182, 8569751709101780459373, 20709480820244837379866, 184792374632521252493091, 544291823175342933160462, 4002137553727571075002992, 15152940733232269015760901, 87018865850686014024398765, 433104886088888993317068254, 1894389925495123040827161995, 13311839736748201810556016501, 460795427538419750098771902028, 949112181811268728834319677752]),
(23, 2, 19, [17578309949348561322559, 20709480820244837043831, 378983175361558116737613, 544291823175342933160462, 8198975842800256794314133, 15152940733232269007631012, 178234094160536697903267279, 433104886088885993065576604, 433104886088887485003249131, 433104886088888993317068254, 3884006626035367269091553784, 13311839736748201810556016501, 84812789010535076185596773292, 460795427538419676011571329897, 460795427538419676011571329898, 460795427538419676112757868566, 460795427538419676129136549709, 460795427538419750098771902028, 20880467999847912034355032910567]),
(23, 3, 16, [12403394925800154563160419, 15152910582375355379254449, 15152910582375360379620865, 15152910582375360379620866, 15152917855617537455824264, 15152939588176843892623664, 15152940733232268810133383, 15152940733232268968553699, 15152940733232269007631012, 433104116334039513582340231, 433104886088885993065576604, 5869620769461733761385694662, 13311839736748201810556016501, 128259789189654551682441132612, 460795427538419676011571329897, 20880467999847912034355032910567]),
(23, 4, 11, [360668496634743777682404399, 433104116334039513582340231, 433104886088885993065576604, 13311295475075886503887408062, 13311323966016870145324106557, 13311839736748201810556002578, 171704715799693544103790853128, 460795407599933748788740451810, 460795413058130180237218933979, 460795427538419676011571329897, 20880467999847912034355032910567]),
(23, 5, 17, [9845043097046282331104781043, 13296706735693447748395554598, 13298153563650460574283291440, 13311295475075883503627451065, 13311295475075883844566315302, 13311295475075886503887408037, 13311295475075886503887408038, 13311509170192171776372102250, 13311839736748201810556002578, 215059953619517785436389470944, 460795407599886256235099156091, 460795407599920416827216102793, 460795407599933748783929453134, 460795427537274691668388776861, 460795427537586816505678303260, 460795427538419676011571329897, 20880467999847912034355032910567]),
(23, 6, 19, [11830651372936896727808467704, 13296706735693447748395554598, 13302164709196952696883050214, 13311295444972519143632576367, 13311295461588553276835594977, 13311295475075883503627451065, 13311839736748201810556002578, 258505201919314757500175095597, 460362866914003179417186254099, 460381589485363117552490009889, 460795406828986351325635905800, 460795406828986414674973530120, 460795406828986422412379358763, 460795406926085239675067611420, 460795407599886256230295937346, 460795407599886256230295937347, 460795407599886256232781614811, 460795407599886256235099142193, 20880467999847912034355032910567]),
(23, 7, 10, [301952210350272621573126154847, 460362866914003179417186254099, 460472681978537543293527899923, 460795406828986348442743684859, 460795406828986350284583831170, 460795406828986351325635905223, 460795407598741197804921658417, 460795407598828685650293888029, 460795407599886256230295937346, 20880467999847912034355032910567]),
(23, 8, 14, [345399211267501728294294046196, 447484112124810369731417374995, 447484112130980457620336870868, 447484112153816171332289970038, 447484124243689509619093729290, 447484132063343792508141040451, 447497982522864589622267937766, 447498720772575371349744560330, 448007887193266745371939779455, 460362323422133206021517451516, 460362380847932393939447227272, 460362866914003179417178124235, 460795406828986348442743684282, 20880467999847912034355032910567]),
(23, 9, 15, [447484112124810369731417361072, 447484112149678402923387615951, 447484112153816171332289970038, 460362322651185805680331682664, 460362322797527496940775984373, 460362323422085710467624328112, 460362323422112203467506312085, 460362323422133203021060332348, 460362566044636107152837023252, 460362866914003105447550915653, 460362866914003120560041826241, 460362866914003179412375255448, 460362866914003179417178124235, 460795406828986348442743684282, 20880467999847912034355032910567]),
(23, 10, 17, [432201811184059864498231164370, 447484112124810369731417361072, 460362322651185805680331682664, 460362323197722259644006589880, 460362323422085710467624328112, 460362759851027013634509738466, 460362846975469685548710638798, 460362848144853183932226718397, 460362866912810625550680852231, 460362866912810626237976850416, 460362866912810628550932344458, 460362866914003105447550915653, 460742261700720797676722892276, 460780274597686517707388052578, 460783741221713715248240130390, 460795406828986348442743670359, 20880467999847912034355032910567]),
(23, 11, 1, [20880467999847912034355032910567]),
(23, 12, 18, [20419672593018925685912289240208, 20419684258626198319106792780173, 20419687725250225516647644857989, 20419725738147191236678310018287, 20420105132933908928907481994914, 20420105132934394132170947551616, 20420105132935101405804100566109, 20420105132935101408117056060147, 20420105132935101408804352058336, 20420105151703058850422806192167, 20420105152872442348806322271769, 20420105239996885020720523172097, 20420105676425826323887408582455, 20420105676650189774711026320684, 20420105677196726228674701227903, 20432983887723101664623615549495, 20448266188663852169856801746193, 20880467999847912034355032910567]),
(23, 13, 17, [20419672593018925685912289226285, 20420105132933908854937854786332, 20420105132933908854938615485664, 20420105132933908854942657655119, 20420105132933908854942657655120, 20420105132933908913794991084322, 20420105132933908928907481994914, 20420105433803275927202195887311, 20420105676425778831333972578219, 20420105676425799830887526598478, 20420105676425826323887408582455, 20420105677050384537414256926190, 20420105677196726228674701227903, 20432983887694095863022742940529, 20432983887698233631431645294612, 20432983887723101664623615549495, 20880467999847912034355032910567]),
(23, 14, 13, [20419672593018925685912289226285, 20420105132933908854937854786332, 20420105676425778828333515459051, 20432969279075336663005288350237, 20432970017325047444732764972797, 20432983867784568241846891870116, 20432983875604222524735939181273, 20432983887694095863022742940529, 20432983887716931576734696039695, 20432983887723101664623615535572, 20432983887723101664623615535573, 20535068788580410306060738864368, 20880467999847912034355032910567]),
(23, 15, 9, [20419672592248025778124736973221, 20419672592249083348704739022534, 20419672592249170836550111252150, 20419672593018925683029397005344, 20419672593018925684070449079393, 20419672593018925685912289225708, 20420105132933908854937846656468, 20578515789497639412781906755716, 20880467999847912034355032910567]),
(23, 16, 18, [20419672592248025778119933768374, 20419672592248025778122251295752, 20419672592248025778124736973221, 20419672592921826794679965299144, 20419672593018925611942653551804, 20419672593018925619680059380443, 20419672593018925683029397004767, 20420086410362548916802542900674, 20420105132933908854937846656468, 20621962797928597276854857814966, 20867156160111163832544476907989, 20867156704372836150851405459502, 20867156704386323481078197315586, 20867156704402939515211400334200, 20867165835138715081658149860350, 20867171293112218586606637355969, 20868637348474975137627224442859, 20880467999847912034355032910567]),
(23, 17, 15, [20419672572309492358343461580670, 20419672572310325217849354607303, 20419672572310637342686644133706, 20419672592247978285571103457433, 20419672592247991617527816807770, 20419672592248025778119933754476, 20665408046228394248918643439620, 20867156160111163832544476907989, 20867156490677719862578660808313, 20867156704372836147851145502530, 20867156704372836150510466595262, 20867156704372836150851405459502, 20867169846284261573780749619124, 20867171293112218586606637355969, 20880467999847912034355032910567]),
(23, 18, 13, [20419672572309492358343461580670, 20419672586789781854117813976584, 20419672592247978285566292458757, 20708763284048218490251242057436, 20867156160111163832544476907989, 20867156675881895164209708804006, 20867156704372836147851145502505, 20867156704372836147851145502506, 20880034894961823148361967333963, 20880034895283514502493530752298, 20880034895731577994841450570336, 20880107331351277290577350506164, 20880467999847912034355032910567]),
(23, 19, 17, [20419672572309492358343461580670, 20752208210658257482672591777952, 20867156160111163832544476894066, 20874598379078450300593647215902, 20880034894961823148361967333963, 20880034895683898795949484788139, 20880034895731577994841450570336, 20880452846907178802086025279555, 20880452846907178802086064356864, 20880452846907178802086222777184, 20880452846907753334039544035331, 20880452846908323857511140286903, 20880452846930056416817577086299, 20880452846937329658994653289702, 20880452846937329658999653656118, 20880455596452986234200469750145, 20880467999847912034355032910567]),
(23, 20, 18, [20419672572309492284256261008539, 20419672572309492340282077148344, 20419672572309492358225896360858, 20419672572309492358242275041997, 20419672572309492358343461580670, 20795655210837376958169436137272, 20867156160111163832544476894066, 20880034894961823145361715842313, 20880034894961823146870029661432, 20880034894961823148361967333963, 20880289765753751497657129643284, 20880452846907178802086025279555, 20880459800872069234098238596431, 20880467455556088859012099750105, 20880467620864736672796916172950, 20880467979138431214110195866736, 20880467982269602085006471588004, 20880467999847912034355032910567]),
(23, 21, 19, [20419672572309492284256261008539, 20838259328350926376760134700430, 20867156160111163832544476894066, 20878573609922416911314205748568, 20880034894961823145361715842313, 20880380980982061348341008511798, 20880452846907178802086017149666, 20880467455556088859012099750105, 20880467979138431214110195530701, 20880467991278160325253252451190, 20880467999076964634008838645385, 20880467999448422437852567625029, 20880467999817761100349133005623, 20880467999829219523875272845663, 20880467999846719480488321546300, 20880467999847031546165111780111, 20880467999847864538796130940900, 20880467999847870207540771580841, 20880467999847912034355032910567]),
(23, 22, 1, [20880467999847912034355032910567]),
]

"""#### 24"""

table_all_equal_by_base += [
(24, 0, 1, [32038681563209056709427351442469835]),
(24, 1, 8944, [1, 73040694872113105, 73040694872113106, 73040694872113107, 73040694872113108, 73040694872113109, 73040694872113110, 73040694872113111, 73040694872113112, 73040694872113113, 73040694872113114, 73040694872113115, 73040694872113116, 73040694872113117, 73040694872113118, 73040694872113119, 73040694872113120, 73040694872113121, 73040694872113122, 73040694872113123, 73040694872113124, 73040694872113125, 73040694872113126, 73040694872113127, 73040694872113128, 73040694872113152, 73040694872113153, 1606895287186489297, 1606895287186489298, 1606895287186489299, 1606895287186489300, 1606895287186489301, 1606895287186489302, 1606895287186489303, 1606895287186489304, 1606895287186489305, 1606895287186489306, 1606895287186489307, 1606895287186489308, 1606895287186489309, 1606895287186489310, 1606895287186489311, 1606895287186489312, 1606895287186489313, 1606895287186489314, 1606895287186489315, 1606895287186489316, 1606895287186489317, 1606895287186489318, 1606895287186489319, 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1019155806131261559275521, 9029013753763285262401534, 27916588318337525511880704, 27916588318337525511880705, 27916588391378220383993809, 27916588391378220383993810, 27916588391378220383993811, 27916588391378220383993812, 27916588391378220383993813, 27916588391378220383993814, 27916588391378220383993815, 27916588391378220383993816, 27916588391378220383993817, 27916588391378220383993818, 27916588391378220383993819, 27916588391378220383993820, 27916588391378220383993821, 27916588391378220383993822, 27916588391378220383993823, 27916588391378220383993824, 27916588391378220383993825, 27916588391378220383993826, 27916588391378220383993827, 27916588391378220383993828, 27916588391378220383993829, 27916588391378220383993830, 27916588391378220383993831, 27916588391378220383993832, 27916588391378220383993856, 27916588391378220383993857, 27916589925232812698370001, 27916589925232812698370002, 27916589925232812698370003, 27916589925232812698370004, 27916589925232812698370005, 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(24, 23, 1, [1333735776850284124449081472843776]),
]

"""#### 25"""

table_all_equal_by_base += [
(25, 0, 0, []),
(25, 1, 706, [1, 2855533828630999398, 3210228865136718750, 3210228865136718751, 65808773040771484326, 65808773040771484327, 65808773040771484328, 65808773040771484329, 65808773040771484330, 65808773040771484331, 65808773040771484332, 65808773040771484333, 65808773040771484334, 65808773040771484335, 65808773040771484336, 65808773040771484337, 65808773040771484338, 65808773040771484339, 65808773040771484340, 65808773040771484341, 65808773040771484342, 65808773040771484343, 65808773040771484344, 65808773040771484345, 65808773040771484346, 65808773040771484347, 65808773040771484348, 65808773040771484349, 65808773040771484350, 87001645711487180730, 1529301166534423828123, 2371809327748840696355, 35653966970963403146923, 65192580223083496093750, 65192580223083496093751, 65195435756912127093148, 65195790451948632812500, 65195790451948632812501, 65258388996124267578076, 65258388996124267578077, 65258388996124267578078, 65258388996124267578079, 65258388996124267578080, 65258388996124267578081, 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(25, 24, 1, [88817841970012523233890533447265625]),
]

"""#### 26"""

table_all_equal_by_base += [
(26, 0, 1, [160182333966853031081693091544779177187]),
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(26, 25, 1, [6156119580207157310796674288400203776]),
]

"""#### 27"""

table_all_equal_by_base += [
(27, 0, 0, []),
(27, 1, 5077, [1, 210391523336766292750, 210391523336766292751, 210391523336766292752, 210391523336766292753, 210391523336766292754, 210391523336766292755, 210391523336766292756, 210391523336766292757, 210391523336766292758, 210391523336766292759, 210391523336766292760, 210391523336766292761, 210391523336766292762, 210391523336766292763, 210391523336766292764, 210391523336766292765, 210391523336766292766, 210391523336766292767, 210391523336766292768, 210391523336766292769, 210391523336766292770, 210391523336766292771, 210391523336766292772, 210391523336766292773, 210391523336766292774, 210391523336766292775, 210391523336766292776, 234513557008986217593, 5268649111578284548879, 5268649111578284548880, 5268649111578284548881, 5268649111578284548882, 5268649111578284548883, 5268649111578284548884, 5268649111578284548885, 5268649111578284548886, 5268649111578284548887, 5268649111578284548888, 5268649111578284548889, 5268649111578284548890, 5268649111578284548891, 5268649111578284548892, 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(27, 26, 1, [443426488243037769948249630619149892803]),
]

"""#### 28"""

table_all_equal_by_base += [
 (28, 0, 1, [928688890453756699447122559347771300777482]),
(28, 1, 4084, [1, 364118239659885068233, 364118239659885068234, 364118239659885068235, 364118239659885068236, 364118239659885068237, 364118239659885068238, 364118239659885068239, 364118239659885068240, 364118239659885068241, 364118239659885068242, 364118239659885068243, 364118239659885068244, 364118239659885068245, 364118239659885068246, 364118239659885068247, 364118239659885068248, 364118239659885068249, 364118239659885068250, 364118239659885068251, 364118239659885068252, 364118239659885068253, 364118239659885068254, 364118239659885068255, 364118239659885068256, 364118239659885068257, 364118239659885068258, 364118239659885068259, 364118239659885068260, 364118239659885068288, 364118239659885068289, 9467074231157011775433, 9467074231157011775434, 9467074231157011775435, 9467074231157011775436, 9467074231157011775437, 9467074231157011775438, 9467074231157011775439, 9467074231157011775440, 9467074231157011775441, 9467074231157011775442, 9467074231157011775443, 9467074231157011775444, 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33145523113244152796222673585053867442172, 33145523113253374862572728253364605812736]),
]

"""#### 29"""

table_all_equal_by_base += [
(29, 0, 1, [74508769042363852559476397161338769391145562]),
(29, 1, 28, [1, 16640166084733083754217, 18411794441625218989315, 449930273440710452666761, 572409104536138903249035, 12245215485485835247014398, 17875085960416221799180043, 334147268034314931596134130, 561607168314929949402976055, 9148527637714527481617133101, 17766624066519996880035186350, 251248794009137992023528375066, 584264599159924056116730185348, 6921354346679532627186888891053, 18832454691704788216146247671392, 191191150631083304879264247330973, 633056896931676876218278250260450, 5294783250024565659797946114616036, 20969946206089369094716635012293949, 146935163170653737898831380254911260, 730772516140847820205333966993677674, 4082044065613172523114524806197008638, 26873447424478584990595446449600684131, 113597050291801292977360308133613575793, 1026683310906978381422887212575756217497, 3218828738846461095006360305251569134268, 44797229296021574864721653238915931344384, 91703076898614683377208150526107718802980]),
(29, 2, 21, [512326952696463915090083906645, 584264599159924056116730185348, 14112770184554182961504829399379, 18832454691704788216146246854811, 389834954965708943027231822661078, 633056896931676876218278250260450, 10788201204146415513265871754094846, 20969946206089369074817673437851083, 20969946206089369086589240421072527, 20969946206089369094716634268968710, 299221627358491992033459623949214339, 730772516140847820205333966993677674, 26873447424478584990595445767115188239, 1026683310906978381404174196995749707292, 1026683310906978381407686815589185999094, 1026683310906978381422887212575756217497, 6455687226884922368615715012294571062610, 44797229296021574864721043865566704851803, 44797229296021574864721270182073236088045, 44797229296021574864721653238915906725040, 2567686153161211134561828214731016126483469]),
(29, 3, 26, [588218069528121139380061643311730, 633056896931676876218278250260450, 16281628319782293127564983700407688, 20969945645054609884322823743834659, 20969945751906945523494187320720359, 20969946206070957890401559776909357, 20969946206070957890404883088874371, 20969946206070957890422957108469641, 20969946206086402831921699773790588, 20969946206089369074817673437851083, 451681941985919263956755726717755123, 730772516140847820205333966993677674, 12551060278536019530829155648479708261, 26873447406729836009426471023069730894, 26873447406729836009426471023069730895, 26873447406730164103010965110189722973, 26873447406730390007326712779078793989, 26873447407607828419798263518095032263, 26873447424478584990595445767115188239, 349418134943873363488153926413878578170, 1026683310906978381404174196290760188432, 1026683310906978381404174196688560253843, 1026683310906978381404174196995749707292, 9744648344003700674920197033252116768926, 44797229296021574864721043865566704851803, 2567686153161211134561828214731016126483469]),
(29, 4, 17, [730772516140847801492318364506948196, 730772516140847809386271407291472732, 730772516140847820205333966993677674, 26873447406729836009426471023069730894, 26873447416768618318918925762223648680, 26873447424478584989986051680537793519, 26873447424478584989986051680537793520, 26873447424478584989986051681136526753, 26873447424478584989986051681281934438, 26873447424478584990471661016312055386, 26873447424478584990576177503027856414, 467407650800944649689386493233775421805, 1026683310906978381404174196290759372752, 44797229295455076335630348958638048700278, 44797229295634642547268520697856527997637, 44797229296021574864721043865566704851803, 2567686153161211134561828214731016126483469]),
(29, 5, 19, [26873447406729836009426471023069730894, 585244706092127067684186754782973255788, 1026662341527270839538198121939490791957, 1026662341527270839538198121939490791958, 1026662341527270853788532277914273531643, 1026662341527270857949382515950831444929, 1026662341527270857949386932074178480439, 1026662341527270857949402415594122678540, 1026663146376648852840545411767794609233, 1026683310906960524748741157676589286272, 1026683310906962533227907610144265030747, 1026683310906978381404174196290759372752, 16322712250084780392948308243210190724378, 44797229295455076335630348958638048700278, 44797229295895496047649962491826715088400, 44797229296021574310723753649276477059023, 44797229296021574535391016565620523099647, 44797229296021574864721043865566704851803, 2567686153161211134561828214731016126483469]),
(29, 6, 24, [25260074550750945730498295332465200995, 26142674908899325945604369565793917815, 26142691494003927522863616563165090762, 26142693723587393565769598740308986865, 26142693723587393578013193012358569780, 26142693723587393584481962203736917474, 26143038729222737057835959146179650987, 26143307380987926360879607628161392495, 26143307381214681751767255643452021881, 26143307381531658443843515442317121080, 26143307381531658443843515442317121081, 26168976139687083217307215985450619921, 26873447406729836009426471023069730894, 703228926905191352500984619751084639747, 1026662341527270839538198100519630618286, 1026662341527270839538198108701468603359, 1026662341527270839538198121939490791957, 1026683310906960524748741157676589286272, 1026683310906974778664951242542673524523, 1026683310906978381404174196290759372752, 44797229295437327586649179983893259918595, 44797229295439109342694761976899856663960, 44797229295455076335630348958638048700278, 2567686153161211134561828214731016126483469]),
(29, 7, 10, [821218364596940092849419116925800607376, 1026662341527270839538198100519630618286, 1026683310906960524748741157676589286272, 22896388993087853305418607235564883461469, 44797229295437327586649179983893259918595, 44797229295448261682485639625179659908118, 44797229295455076335630329690373961396446, 44797229295455076335630347091427273410984, 44797229295455076335630348958638048700278, 2567686153161211134561828214731016126483469]),
(29, 8, 11, [939050452099568639136688524262717803405, 1026662341527270839538198100519630618286, 1026679435201752824026214261921945765149, 1026683310906960524748741136257497056221, 1026683310906960524748741136257497056222, 26185344815162629076958876586508518670337, 44770355848614862349799677568986920399240, 44771086621112692841452295376061302842059, 44773707752636967012423574181159695661702, 44797229295437327586649179983893259918595, 2567686153161211134561828214731016126483469]),
(29, 9, 18, [29474305937344760539770992883256216131317, 44770355848597096298142262108245812238466, 44770355848597096298142643902186191514071, 44770355848597096298142852234068277709109, 44770355848614862349799677568986920399209, 44770355848614862349799677568986920399210, 44770833994409616373364465689833216069011, 44771085988074105711945715837006231279126, 44771086000523803205013837452156427471769, 44771086006888232744806061299021019312895, 44771086621112692841452295355346456805691, 44771086621112692841452295368561206194630, 44771086621112692841452295376061302842059, 44777942264463499032766914409806511321514, 44797229295437327568237975689533613895641, 44797229295437327580099975671508987800126, 44797229295437327586649179983893259918595, 2567686153161211134561828214731016126483469]),
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(29, 14, 1, [2567686153161211134561828214731016126483469]),
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(29, 24, 15, [2522888923865189559697107170865449421631666, 2522888923865576492014559694033159598485828, 2522888923865756058226197865772378077783191, 2566659469850304156180424040534725367110717, 2567218745510410189912138828237782351061660, 2567659279713786655976837638553513098627055, 2567659279713786655976837743069999814428080, 2567659279713786655976838228679334844549031, 2567659279713786655976838228679334989956712, 2567659279713786655976838228679335588689950, 2567659279713804404725818788259993056752575, 2567685422388694993714008009397049132805795, 2567685422388694993714018828459608835010734, 2567685422388694993714026722412651619535273, 2567686153161211134561828214731016126483469]),
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(29, 26, 22, [2522888923865189559697106561492100219758429, 2522888923865189559697106944548942890395421, 2522888923865189559697107170865449421631666, 2561230465934326212193212499718721555420855, 2566659469850304156180405327518440370265972, 2566659469850304156180420527915426940484372, 2566659469850304156180424040534020376776177, 2567659279713786655976837619285249011295230, 2567677836611742479670520188009414829113700, 2567685422388694993714008009397049132805795, 2567685853939583776069836181271392177269126, 2567686132191264928472459120014381857514759, 2567686132191264928472459128141775705410938, 2567686132191264928472459139913342688632386, 2567686142373009930415412701465144372388619, 2567686152528154237630151338512737876223019, 2567686152771376179596119271703784303822387, 2567686153142378679870123426514869879628658, 2567686153147098364377274031769511297084087, 2567686153160626869962668290674899396298121, 2567686153160698807609131750815926042576820, 2567686153161211134561828214731016126483469]),
(29, 27, 28, [2522888923865189559697106561492100195139085, 2564467324422364673466821854425764557349197, 2566659469850304156180405327518440370265972, 2567659279713786655976837619284566525799338, 2567682071117145521389305100206209929474827, 2567685422388694993714008009397049132805795, 2567686006226047963908090315899635871572206, 2567686132191264928472459120014381114189520, 2567686147866427884537262554933070011867429, 2567686152528154237630151338512737876223019, 2567686152970019983930744909851751879152493, 2567686153142378679870123426514869878812077, 2567686153154289780215148682103829237592413, 2567686153160626869962668290674899396298121, 2567686153160959885767819076738992598108400, 2567686153161193367937761694734136091297119, 2567686153161201986034190500203534509350365, 2567686153161210572954659899801066723507414, 2567686153161211116686742254314794327303426, 2567686153161211122316612729245180879469067, 2567686153161211133989419110194877223234434, 2567686153161211133989419110194877223234435, 2567686153161211133989419110194877223234464, 2567686153161211133989419110194877223234465, 2567686153161211134111897941290305673816704, 2567686153161211134543416420289390907494154, 2567686153161211134545188048646283042729248, 2567686153161211134561828214731016126483469]),
(29, 28, 1, [2567686153161211134561828214731016126483469]),
]

"""#### 30"""

table_all_equal_by_base += [
(30, 0, 0, []),
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(30, 29, 1, [205891132094649000000000000000000000000000000]),
]

"""#### 31"""

table_all_equal_by_base += [
(31, 0, 1, [529428987529739460369842168744635422842585510266]),
(31, 1, 27, [1, 1406226825609742097338778, 1545975126436265392309150, 40831185682357866916239405, 51147375115840602941572426, 1192294458849455786805456485, 1698806330871969470468636803, 34910393220057786629250928492, 56724092037483074943270899875, 1024523674081016702528269712767, 1903722958918693503839510272034, 66283915495180354731732125001826, 891323958925742563209234405511645, 2260520155337880423725505596057569, 26376538067790096346087097385589109, 77859012035160753160169171258654721, 2793020404691190465462267118188222530, 23224516926576221424367550130030390708, 101672077963139664041713273905355859308, 691570897231263044193733101272502928315, 3777926203369590818668983508434586759565, 20604713180235075474539545069739651021109, 144185532396197866876665899220566647979701, 6018012273273555431401556122026376681987568, 18660726714689274915016394097821387966688028, 278320155234894663618967851002563194574008088, 568972471024107865287021434301977158534824480]),
(31, 2, 18, [61415849040976783753028280830494, 66283915495180354731732125001826, 1813444792487771219791583064421343, 2260520155337880423725505596056544, 77859012035160753158977113521611842, 2793020404691190465462267118188222530, 47258683350929543864959719191653131412, 101672077963139664041713237124301187304, 1406394526580749362038815174350364314693, 3777926203369590818668983507297763445662, 144185532396197866875277958749168777684726, 144185532396197866875519794885921425641984, 144185532396197866876665899220566647979701, 6018012273273555431401556082087585851983938, 6018012273273555431401556122026341454344788, 37411849923603251696786581527308799173749801, 278320155234894663618967851002563194574008088, 17069174130723235958610643029059314756044734431]),
(31, 3, 18, [2400674815765812982087719417046076127, 2793020404691190465462267118188222530, 71320156217399849016742558107926860162, 101672077963139664041713237124301187304, 2121999597421697551476440408388106965061, 3777926203367943159713227379603379466803, 3777926203369590818668983507297763445662, 63262594293551227589056633083797981839100, 144185532396197866875277958749168777684726, 1888074417415455737613559509317184374949939, 6018012273273555431401556082087585851983938, 278320155234894606896421787346440044010055286, 278320155234894606946023229791026262666594272, 278320155234894620044123918326486670375380218, 278320155234894663618967851002525241469434386, 278320155234894663618967851002539692844917202, 278320155234894663618967851002563159346365275, 17069174130723235958610643029059314756044734431]),
(31, 4, 26, [95354330262124582431964298917143620851, 101669883671924135689538443619728930385, 101670328381173073949337054001359582075, 101672077963139664040453729396642565351, 2837631967404131438901322226056829802885, 3777926203367943159713227379567059694163, 3777926203367943159713227379579387001361, 3777926203367943159713227379603379466803, 84590622727963912239517867863503074583665, 144185532396197866875277918810377948736081, 2525024420723989931252136499525173269133426, 6018012195414545093502547948538338792327532, 6018012195414545093502579045539613178539486, 6018012195414545093502590363743511192668657, 6018012273207271565507818211879691196540035, 6018012273207271565508360998838753333771913, 6018012273207271565509206152352261116670943, 6018012273208478753753513944626905094760196, 6018012273209175238865293161350792469451842, 6018012273234501642063610348900559226667153, 6018012273273555431401556082087585851983905, 6018012273273555431401556082087585851983906, 278320155234894606896421787346440044010055286, 278320155234894656148267100587898355266516753, 278320155234894663618967851002525241468379368, 17069174130723235958610643029059314756044734431]),
(31, 5, 11, [3777926203367943159713227379567059694163, 144185532396197866875277918810377948736081, 3161951164265875698118727576231763299378597, 6018012195414545093502547948501522511067733, 6018012224876651230632355806389709982418186, 6018012273207271565507818211879691196540035, 6018012273273555429855623410832064777609610, 6018012273273555431073483823688639080347494, 6018012273273555431401556082087585851951039, 278320155234894606896421787346440044010055286, 17069174130723235958610643029059314756044734431]),
(31, 6, 8, [127232124376602015998790930819570331753897, 144185532396197866875277918810377948736081, 3798185624168020973570178383010555092546295, 6018012195414545093502547948501522511067733, 6018012252176093974565864801079400736166258, 6018012273207271565507818211879691196507136, 278320155234894606896421787346440044010055286, 17069174130723235958610643029059314756044734431]),
(31, 7, 21, [4435135627507810265711011519487290973440387, 6018012195414545093502547948500385687753830, 6018012195414545093502547948501207966987648, 6018012195414545093502547948501522511067733, 278316377386484022149360727390744170145263284, 278316377386485412092659603803836745281857284, 278316377386485925821170882987996081191472159, 278316377386550306013707145948671630210270338, 278316379394910790409709624487827683041213640, 278316380099451123589862046362315525648053083, 278316421069880303072111733116089631130455588, 278316479056367692376236018727281747515245535, 278320155232634086792231282078493932184518788, 278320155232634086792231288718050115108127496, 278320155232634086792231324493699069357217133, 278320155232728661411993735701649668903310327, 278320155234894606894875854675185659758994894, 278320155234894606894875855601318774585975676, 278320155234894606894875855935922262529104403, 278320155234894606896421787346440044010055286, 17069174130723235958610643029059314756044734431]),
(31, 8, 21, [5873826743071648828108752132183314324166491, 6018012195414543396242233318725868656510712, 6018012195414544406074104067728259013463768, 6018012195414545093501202423270941271449865, 6018012195414545093502347114413638299991573, 6018012195414545093502547948500384595623970, 151601607703519890358561539299170049398202958, 278316377386484022149360727390744170145263284, 278316377386539544214541480752150365818257109, 278316377386550249340763870222463341314559176, 278316377386550249340763870238605855836453049, 278316377386550249340763870262402131016382794, 278316377386550306013707145948671630210270338, 278316445104976152624270528872533092567836810, 278316478978508681988735610564351454198369149, 278316479056367692376236018727281747515245535, 278317661572283908620257734251702698083289871, 278320155232634086792231282077301839219832104, 278320155233620499265482601361699979325703188, 278320155234894606894875854675185659758962028, 17069174130723235958610643029059314756044734431]),
(31, 9, 18, [5708320055162447852970427718684705333863734, 5873826743071648828108752132183314324166491, 6018012195414543396242233318725868656510712, 278316377386484022149360727390707389056651131, 278316469139149532876045651094857003656060671, 278316476263413573197052315519541253226603419, 278316476263413573202865546827413584839102522, 278316476263413573246653715548885519487406838, 278316476367638354871613852793952420906941921, 278316478978508681988735609264904934618636850, 278316479011153888570734984256882521031504685, 278316479056367692374690087316764001226942232, 278316479056367692375719681543443068285014267, 278316479056367692376236018727281747481339244, 278320155232634086792231282077301839219832104, 278320155234542566896629448909114920862504961, 278320155234894606894875854675185659758962028, 17069174130723235958610643029059314756044734431]),
(31, 10, 5, [189672444419800040845893346406321200852477729, 278316377386484022149360727390707389056651131, 278319092713158440563201512061939600924973878, 278320155232634086792231282077301839219832104, 17069174130723235958610643029059314756044734431]),
(31, 11, 21, [208707517382319837521641134121536204595426434, 272302143037219600068933711126964487070457359, 272302143037219600068933750179677375973876251, 272302143037219600068933753542206439432284618, 272302143037272192704048939990079259105296102, 272302143037285827260336896413864365110723649, 272302143037856362662573179329154686964172121, 272302143039480061852520640726729271054712932, 272302143039480118525465303054024855781680556, 272369805098018165084570925001335385045869874, 272446328491823013093598784240567373795308147, 278316377386484022149360727350805414508963277, 278316377386484022149360727390707389056651131, 278319807633345180731509066411033888535376276, 278320053562816700429702409259945024050563105, 278320055258951184334868009317141568591353040, 278320056277911809221384316364282654008947317, 278320155232634086790685350667976150669627723, 278320155232634086791620782895900220252367670, 278320155232634086792231282077301839219832104, 17069174130723235958610643029059314756044734431]),
(31, 12, 15, [227721245182880336345149536888779370534563587, 272302143037219600068933711126964485944354484, 272302143038747692705109729897613395902526805, 272302143039480061852520640726729271054712932, 272391125470837882699555678483820971224851423, 272446328491823013093598784240567373795308147, 273409797527154059511598002141151670939409691, 278316377308691293978547780730167122895710771, 278316377308691294016522659066412698573690314, 278316377308691294028149223174752168408141062, 278316377308691294028149224324425278417613051, 278316377308691294028149224435489943229174388, 278316377318076595201832512520694847488043684, 278316377386484022149360727350805414508963277, 17069174130723235958610643029059314756044734431]),
(31, 13, 27, [246756317337453353318323275843835545471663084, 272302143037219543395989091174973235927293191, 272302143037219546235991716969621083454975507, 272302143037219600068933711126964485944354484, 272412446626523587592656827370278022969605230, 272442550643412373321250587278150667122942115, 272442550643414185408317962527830261813457864, 272442550643414276993062130855813903691641723, 272442550643478655537939480435663783035005652, 272442550643478656820766109833274720109991642, 272442550643478657185598393776549525057156194, 272442614866671687815123623609948918203651947, 272442652313296043546581335184580686902808607, 272446328491823013092052810413615582181545919, 272446328491823013092894082719860881069572392, 272446328491823013093598741824133363696435959, 272446328491823013093598741824133363696435960, 272446328491823013093598741824161254651486890, 272446328491823013093598741824170144750052945, 274046747557761720438408133221242959568464313, 278316377308691293978547780730165949751569210, 278316377386484022149359339410333981412113406, 278316377386484022149359739704063853287400890, 278316377386484022149360727350805379316315664, 278316377386484022149360727350805390731471308, 278316377386484022149360727350805414508963277, 17069174130723235958610643029059314756044734431]),
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(31, 15, 1, [17069174130723235958610643029059314756044734431]),
(31, 16, 25, [16790857753336751936461283689648980774632621025, 16790857753336751936483182803704513903299078276, 16790857753336751936512429676824349943350185654, 16790857753338756026910465795538489536022851873, 16790857753338946229374071050745823614343073761, 16790857753351483549150222533015071905770811848, 16790857753414544664632095248329148806293165221, 16794491148500595981181400323528129050834923405, 16796727802231412945518590218645699173863188512, 16796731478409939915064061693875964079983543430, 16796731480601317549802101785719147072904123662, 16796731481200699742976571060291302917306681512, 16796731491822390890980830671016512999156762065, 16796731580079757303072703548623650973009728812, 16796731580079757303072703577715971240163253790, 16796731580079757303072703590998955308811763978, 16796731580079757303072703591035113639092689108, 16796731580079757303072703591038856145444009061, 16796731580079773864179115584925116280411167746, 16796731580079823585289392441781164088922848359, 16796740363750741824762673401995729267855626225, 16796871987686016358541709317932350270100379947, 16796871987686016377462851842280653457874393210, 16796871987686016415214653937884341520117441240, 17069174130723235958610643029059314756044734431]),
(31, 17, 26, [16790857753336751936461282301708509341535771154, 16790857753336751936461282301708509365313263119, 16790857753336751936461282301708509376728418767, 16790857753336751936461283289355250902757333537, 16790857753336751936461283689648980774632621025, 16790857753414544664632095248329148806293165221, 16795127383165474238172234895838071796476270114, 16796727802231412945517044287235144611294681486, 16796727802231412945517044287235153501393247537, 16796727802231412945517044287235181392348298472, 16796727802231412945517748946339453874975162035, 16796727802231412945518590218645699173863188512, 16796731478409939915064061693874734069141925824, 16796731580079757301425044635282765230987578237, 16796731580079757301789876919226040035934742785, 16796731580079757303072703548623650973009728779, 16796731580079757303072703548623650973009728780, 16796731580079821681617580898203500852353092708, 16796731580079821773202325066531484494231276563, 16796731580079823585289392441781164088921792316, 16796761684096712371017986201689036733075129198, 16796871987686016358541709317932350270100379947, 16796871987686016412374651312089693672589758920, 16796871987686016415214653937884341520117441240, 16822417813385782605292319753215479210573071344, 17069174130723235958610643029059314756044734431]),
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(31, 23, 22, [16790853975488341351714221241712874712034679145, 16790853975488341351715767173123392493515630028, 16790853975488341351715767173123392493515630029, 16790853975488341351715767173457995981458758751, 16790853975488341351715767174384129096285739537, 16790853975490507297198649293357665087141424100, 16790853975490601871818411704565615686687517298, 16790853975490601871818411746980820823860215643, 16790857651666868266234407010332033008529488896, 16790857709653355655538531295943225124914278840, 16790857750623784835020780982696999230396681348, 16790857751328325168200933404571487073003520787, 16790857753336685652596935883110643125834464093, 16790857753336750032789472146071318674853262272, 16790857753336750546517983425255478010762877143, 16790857753336751936461282301668570585899471147, 16936607595199890590664196198831634047757777946, 17063156118527821413517140481110813233533666698, 17063156118527821413517140481110813548077746779, 17063156118527821413517140481110814370356980601, 17064738995095728148344932017539827465071294040, 17069174130723235958610643029059314756044734431]),
(31, 24, 8, [16790853975488341351714221241712874712034679145, 17063156118450028687045135210847435064848227295, 17063156118471059864636077164258235355308568169, 17063156118527821413517140481110813233533666698, 17065375945099067937637072850676304200952188132, 17069029945190839760743767751140504378095998350, 17069046898598859356594644238128495185712980531, 17069174130723235958610643029059314756044734431]),
(31, 25, 12, [16790853975488341351714221241712874712034679145, 17063156118449962403179241472977227170192783392, 17063156118449962403179569545235626116964386933, 17063156118449962403180787405648482691267124821, 17063156118449970200677323244003869805313009605, 17063156118450028687045135210847435064848194396, 17063156118498359307380010673252925046062316241, 17063156118527821413517140481110813233533666698, 17066012179558970082912524301483082992745355830, 17069029945190839760743767751140504378095998350, 17069170352797032590667483315831935188985040268, 17069174130723235958610643029059314756044734431]),
(31, 26, 25, [16790853975488341294991675178056789514576355063, 16790853975488341302462375928471416400778217674, 16790853975488341351714221241712874712034679145, 17063156118449962403179241472977227170192750526, 17063156118450001456968579418710414196818067274, 17063156118450026783371777735897963963575282589, 17063156118450027479856889515114687850949974231, 17063156118450028687045133822906962494928063488, 17063156118450028687045134668060476002710962515, 17063156118450028687045135210847435064848194396, 17063156118527821413517140438695571244852065774, 17063156118527821413517140450013775142866194941, 17063156118527821413517140481110776417252406899, 17066649106302511968679390892559789582775601001, 17069029945190839760743767751140504378095998350, 17069170352797032590667483315831935152665267628, 17069170352797032590667483315831935176657733066, 17069170352797032590667483315831935188985040268, 17069171293091268554479204127737088699214931542, 17069174029051157995470978988605585359402169080, 17069174029051157995470978988605585359402169081, 17069174029052907577437569079722260754685152352, 17069174029053352286686507339520871136315804046, 17069174035368905696486060597095015838901113576, 17069174130723235958610643029059314756044734431]),
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(31, 29, 26, [16790853975488341294991675178056751561470726343, 17050513404008546683695626634961493368078046400, 17063156118449962403179241472937288379362746863, 17069029945190839760743766363160094189396754730, 17069153526010055723535168489514245016393713318, 17069170352797032589019824360075806321457974866, 17069173439152338727347598835326213483541806112, 17069174029051157995470978987346040850688875123, 17069174107498719032034421604691764626014343719, 17069174127930215553919452563597047637856511901, 17069174130645376946575482275899145584786079710, 17069174130696859420542852932713227658659145319, 17069174130720975438455305148635589250448676862, 17069174130722344634651717286496105521639222782, 17069174130723169674695147848704583023919732605, 17069174130723234054887684110365810916534462397, 17069174130723234934086968948042612227775021661, 17069174130723235901886550991576239812773834556, 17069174130723235923700249809001528126793805935, 17069174130723235956911836698187345285576097628, 17069174130723235957418348570209858969239277943, 17069174130723235958559495653943474153103162005, 17069174130723235958569811843376956889128495022, 17069174130723235958609097053932878490652425281, 17069174130723235958609236802233705013947395649, 17069174130723235958610643029059314756044734431]),
(31, 30, 1, [17069174130723235958610643029059314756044734431]),
]

"""#### 32"""

table_all_equal_by_base += [
(32, 0, 0, []),
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1461501637330901064431951393184302652598190604276, 1461501637330901064431951393184302652598190604277, 1461501637330901064431951393184302652598190604278, 1461501637330901064431951393184302652598190604279, 1461501637330901064431951393184302652598190604280, 1461501637330901064431951393184302652598190604281, 1461501637330901064431951393184302652598190604282, 1461501637330901064431951393184302652598190604283, 1461501637330901064431951393184302652598190604284, 1461501637330901064431951393184302652598190604285, 1461501637330902799822614829431447679131870822400, 1461501637330902857356456021761271747814178684895, 1461501637330902857356456021761271747814178684896, 1461501637330902857356456021761271747814178684897, 1461501637330902857356456021761271747814178684898, 1461501637330902857356456021761271747814178684899, 1461501637330902857356456021761271747814178684900, 1461501637330902857356456021761271747814178684901, 1461501637330902857356456021761271747814178684902, 1461501637330902857356456021761271747814178684903, 1461501637330902857356456021761271747814178684904, 1461501637330902857356456021761271747814178684905, 1461501637330902857356456021761271747814178684906, 1461501637330902857356456021761271747814178684907, 1461501637330902857356456021761271747814178684908, 1461501637330902857356456021761271747814178684909, 1461501637330902857356456021761271747814178684910, 1461501637330902857356456021761271747814178684911, 1461501637330902857356456021761271747814178684912, 1461501637330902857356456021761271747814178684913, 1461501637330902857356456021761271747814178684914, 1461501637330902857356456021761271747814178684915, 1461501637330902857356456021761271747814178684916, 1461501637330902857356456021761271747814178684917, 1461501637330902857356456021761271747814178684918, 1461501637330902857356456021761271747814178684919, 1461501637330902857356456021761271747814178684920, 1461501637330902857356456021761271747814178684921, 1461501637330902857356456021761271747814178684922, 1461501637330902857356456021761271747814178684923, 1461501637330902857356456021761271747814178684924, 1461501637330902857356456021761271747814178684925, 1461501637330902914788372678239386430777008848896, 1461501637330902916203173729231108495418947796959, 1461501637330902916203173729231108495418947796960, 1461501637330902916203173729231108495418947796961, 1461501637330902916203173729231108495418947796962, 1461501637330902916203173729231108495418947796963, 1461501637330902916203173729231108495418947796964, 1461501637330902916203173729231108495418947796965, 1461501637330902916203173729231108495418947796966, 1461501637330902916203173729231108495418947796967, 1461501637330902916203173729231108495418947796968, 1461501637330902916203173729231108495418947796969, 1461501637330902916203173729231108495418947796970, 1461501637330902916203173729231108495418947796971, 1461501637330902916203173729231108495418947796972, 1461501637330902916203173729231108495418947796973, 1461501637330902916203173729231108495418947796974, 1461501637330902916203173729231108495418947796975, 1461501637330902916203173729231108495418947796976, 1461501637330902916203173729231108495418947796977, 1461501637330902916203173729231108495418947796978, 1461501637330902916203173729231108495418947796979, 1461501637330902916203173729231108495418947796980, 1461501637330902916203173729231108495418947796981, 1461501637330902916203173729231108495418947796982, 1461501637330902916203173729231108495418947796983, 1461501637330902916203173729231108495418947796984, 1461501637330902916203173729231108495418947796985, 1461501637330902916203173729231108495418947796986, 1461501637330902916203173729231108495418947796987, 1461501637330902916203173729231108495418947796988, 1461501637330902916203173729231108495418947796989, 1461501637330902918104565004766079573621773172736, 1461501637330902918104577094024275719913520234463, 1461501637330902918104577094024275719913520234464, 1461501637330902918104577094024275719913520234465, 1461501637330902918104577094024275719913520234466, 1461501637330902918104577094024275719913520234467, 1461501637330902918104577094024275719913520234468, 1461501637330902918104577094024275719913520234469, 1461501637330902918104577094024275719913520234470, 1461501637330902918104577094024275719913520234471, 1461501637330902918104577094024275719913520234472, 1461501637330902918104577094024275719913520234473, 1461501637330902918104577094024275719913520234474, 1461501637330902918104577094024275719913520234475, 1461501637330902918104577094024275719913520234476, 1461501637330902918104577094024275719913520234477, 1461501637330902918104577094024275719913520234478, 1461501637330902918104577094024275719913520234479, 1461501637330902918104577094024275719913520234480, 1461501637330902918104577094024275719913520234481, 1461501637330902918104577094024275719913520234482, 1461501637330902918104577094024275719913520234483, 1461501637330902918104577094024275719913520234484, 1461501637330902918104577094024275719913520234485, 1461501637330902918104577094024275719913520234486, 1461501637330902918104577094024275719913520234487, 1461501637330902918104577094024275719913520234488, 1461501637330902918104577094024275719913520234489, 1461501637330902918104577094024275719913520234490, 1461501637330902918104577094024275719913520234491, 1461501637330902918104577094024275719913520234492, 1461501637330902918104577094024275719913520234493, 1461501637330902918104649629573452597664002605056, 1461501637330902918137711332888273476333867106271, 1461501637330902918137711332888273476333867106272, 1461501637330902918137711332888273476333867106273, 1461501637330902918137711332888273476333867106274, 1461501637330902918137711332888273476333867106275, 1461501637330902918137711332888273476333867106276, 1461501637330902918137711332888273476333867106277, 1461501637330902918137711332888273476333867106278, 1461501637330902918137711332888273476333867106279, 1461501637330902918137711332888273476333867106280, 1461501637330902918137711332888273476333867106281, 1461501637330902918137711332888273476333867106282, 1461501637330902918137711332888273476333867106283, 1461501637330902918137711332888273476333867106284, 1461501637330902918137711332888273476333867106285, 1461501637330902918137711332888273476333867106286, 1461501637330902918137711332888273476333867106287, 1461501637330902918137711332888273476333867106288, 1461501637330902918137711332888273476333867106289, 1461501637330902918137711332888273476333867106290, 1461501637330902918137711332888273476333867106291, 1461501637330902918137711332888273476333867106292, 1461501637330902918137711332888273476333867106293, 1461501637330902918137711332888273476333867106294, 1461501637330902918137711332888273476333867106295, 1461501637330902918137711332888273476333867106296, 1461501637330902918137711332888273476333867106297, 1461501637330902918137711332888273476333867106298, 1461501637330902918137711332888273476333867106299, 1461501637330902918137711332888273476333867106300, 1461501637330902918137711332888273476333867106301, 1461501637330902918200795500007404055928384782336, 1461501637330902918201501512686058999366413189087, 1461501637330902918201501512686058999366413189088, 1461501637330902918201501512686058999366413189089, 1461501637330902918201501512686058999366413189090, 1461501637330902918201501512686058999366413189091, 1461501637330902918201501512686058999366413189092, 1461501637330902918201501512686058999366413189093, 1461501637330902918201501512686058999366413189094, 1461501637330902918201501512686058999366413189095, 1461501637330902918201501512686058999366413189096, 1461501637330902918201501512686058999366413189097, 1461501637330902918201501512686058999366413189098, 1461501637330902918201501512686058999366413189099, 1461501637330902918201501512686058999366413189100, 1461501637330902918201501512686058999366413189101, 1461501637330902918201501512686058999366413189102, 1461501637330902918201501512686058999366413189103, 1461501637330902918201501512686058999366413189104, 1461501637330902918201501512686058999366413189105, 1461501637330902918201501512686058999366413189106, 1461501637330902918201501512686058999366413189107, 1461501637330902918201501512686058999366413189108, 1461501637330902918201501512686058999366413189109, 1461501637330902918201501512686058999366413189110, 1461501637330902918201501512686058999366413189111, 1461501637330902918201501512686058999366413189112, 1461501637330902918201501512686058999366413189113, 1461501637330902918201501512686058999366413189114, 1461501637330902918201501512686058999366413189115, 1461501637330902918201501512686058999366413189116, 1461501637330902918201501512686058999366413189117, 1461501637330902918203600207908909995613703110656, 1461501637330902918203612297167106141905450172383, 1461501637330902918203612297167106141905450172384, 1461501637330902918203612297167106141905450172385, 1461501637330902918203612297167106141905450172386, 1461501637330902918203612297167106141905450172387, 1461501637330902918203612297167106141905450172388, 1461501637330902918203612297167106141905450172389, 1461501637330902918203612297167106141905450172390, 1461501637330902918203612297167106141905450172391, 1461501637330902918203612297167106141905450172392, 1461501637330902918203612297167106141905450172393, 1461501637330902918203612297167106141905450172394, 1461501637330902918203612297167106141905450172395, 1461501637330902918203612297167106141905450172396, 1461501637330902918203612297167106141905450172397, 1461501637330902918203612297167106141905450172398, 1461501637330902918203612297167106141905450172399, 1461501637330902918203612297167106141905450172400, 1461501637330902918203612297167106141905450172401, 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(32, 31, 1, [1461501637330902918203684832716283019655932542976]),


                             ]

"""#### 33"""

table_all_equal_by_base += [
(33, 0, 1, [4262649311868962034947877223846561239424294726563632]),
(33, 1, 10423, [1, 126458318455345821115073464, 126458318455345821115073465, 126458318455345821115073466, 126458318455345821115073467, 126458318455345821115073468, 126458318455345821115073469, 126458318455345821115073470, 126458318455345821115073471, 126458318455345821115073472, 126458318455345821115073473, 126458318455345821115073474, 126458318455345821115073475, 126458318455345821115073476, 126458318455345821115073477, 126458318455345821115073478, 126458318455345821115073479, 126458318455345821115073480, 126458318455345821115073481, 126458318455345821115073482, 126458318455345821115073483, 126458318455345821115073484, 126458318455345821115073485, 126458318455345821115073486, 126458318455345821115073487, 126458318455345821115073488, 126458318455345821115073489, 126458318455345821115073490, 126458318455345821115073491, 126458318455345821115073492, 126458318455345821115073493, 126458318455345821115073494, 126458318455345821115073495, 126458318455345821115073496, 138241368658610922837305943, 138241368658610922837305944, 3924451039598378037168371557, 3924451039598378037168371558, 3924451039598378037168371559, 3924451039598378037168371560, 3924451039598378037168371561, 3924451039598378037168371562, 3924451039598378037168371563, 3924451039598378037168371564, 3924451039598378037168371565, 3924451039598378037168371566, 3924451039598378037168371567, 3924451039598378037168371568, 3924451039598378037168371569, 3924451039598378037168371570, 3924451039598378037168371571, 3924451039598378037168371572, 3924451039598378037168371573, 3924451039598378037168371574, 3924451039598378037168371575, 3924451039598378037168371576, 3924451039598378037168371577, 3924451039598378037168371578, 3924451039598378037168371579, 3924451039598378037168371580, 3924451039598378037168371581, 3924451039598378037168371582, 3924451039598378037168371583, 3924451039598378037168371584, 3924451039598378037168371585, 3924451039598378037168371586, 3924451039598378037168371587, 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129110040087761020172275875758366688195614533348383, 129110040087761020172275875758366688195614533348384, 129110040087761020172275875758366688195614533348385, 129110040087761020172275875758366688195614533348386, 129110040087761020172275875758366688195614533348387, 129110040087761020172275875758366688195614533348388, 129110040087761020172275875758366688195614533348389, 129110040087761020172275875758366688195614533348390, 129110040087761020172275875758366688195614533348391, 129110040087761020172275875758366688195614533348392, 129110040087761020172275875758366688195614533348393, 129110040087761020172275875758366688195614533348394, 129110040087761020172275875758366688195614533348395, 129110040087761020172275875758366688195614533348396, 129110040087761020172275875758366688195614533348397, 129110040087761020172275875758366688195614533348398, 129110040087761020175931727936025875180851071044734, 129110040087761020175980041525263106311966638372895, 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129110040087761027719738790366249923382029881291922, 129110040087761027719738790366249923382029881291923, 129110040087761027719738790366249923382029881291924, 129110040087761027719738790366249923382029881291925, 129110040087761027719738790366249923382029881291926, 129110040087761027719738790366249923382029881291927, 129110040087761027833583409680679700701918961909491, 129110040087761027835789468293086919080954048202834, 129110040087761027835789468293086919080954048202835, 129110040087761027835789468293086919080954048202836, 129110040087761027835789468293086919080954048202837, 129110040087761027835789468293086919080954048202838, 129110040087761027835789468293086919080954048202839, 129110040087761027835789468293086919080954048202840, 129110040087761027835789468293086919080954048202841, 129110040087761027835789468293086919080954048202842, 129110040087761027835789468293086919080954048202843, 129110040087761027835789468293086919080954048202844, 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(33, 32, 1, [129110040087761027839616029934664535539337183380513]),

                             ]

"""#### 34"""

table_all_equal_by_base += [
(34, 0, 1, [399903937958473433782862763628747974628490691628136485]),
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(34, 33, 1, [11756638905368616011414050501310355554617941909569536]),

                             ]

"""#### 35"""

table_all_equal_by_base += [
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1102507499354148695951347082928830249429165070195312467, 1102507499354148695951347082928830249429165070195312468, 1102507499354148695951347082928830249429165070195312469, 1102507499354148695951347082928830249429165070195312470, 1102507499354148695951347082928830249429165070195312471, 1102507499354148695951347082928830249429165070195312472, 1102507499354148695951347082928830249429165070195312473, 1102507499354148695951347082928830249429165070195312474, 1102507499354148695951347082928830249429165070195312475, 1102507499354148695951347082928830249429165070195312476, 1102507499354148695951347082928830249429165070195312477, 1102507499354148695951347082928830249429165070195312478, 1102507499354148695951347082928830249429165070195312479, 1102507499354148695951347082928830249429165070195312480, 1102507499354148695951347082928830249429165070195312481, 1102507499354148695951347082928830249429165070195312482, 1102507499354148695951347082928830249429165070195312483, 1102507499354148695951347082928830249429165070195312484, 1102507499354148695951347082928830249429165070195312485, 1102507499354148695951347082928830249429165070195312486, 1102507499354148695951347082928830249429165070195312487, 1102507499354148695951347082928830249429165070195312488, 1102507499354148695951347082928830249429165070195312489, 1102507499354148695951347082928830249429165070195312490, 1102507499354148695951347082928830249429165070195312491, 1102507499354148695951347082928830249429165070195312492, 1102507499354148695951347082928830249429165070195312493, 1102507499354148695951347082928830249429165070195312494, 1102507499354148695951347082928830249429165070195312495, 1102507499354148695951347082928830249429165070195312496, 1102507499354148695951347082928830249429165070195312497, 1102507499354148695951768349175752279345020970879189853, 1102507499354148695951773219604919180708919708867187464, 1102507499354148695951773219604919180708919708867187465, 1102507499354148695951773219604919180708919708867187466, 1102507499354148695951773219604919180708919708867187467, 1102507499354148695951773219604919180708919708867187468, 1102507499354148695951773219604919180708919708867187469, 1102507499354148695951773219604919180708919708867187470, 1102507499354148695951773219604919180708919708867187471, 1102507499354148695951773219604919180708919708867187472, 1102507499354148695951773219604919180708919708867187473, 1102507499354148695951773219604919180708919708867187474, 1102507499354148695951773219604919180708919708867187475, 1102507499354148695951773219604919180708919708867187476, 1102507499354148695951773219604919180708919708867187477, 1102507499354148695951773219604919180708919708867187478, 1102507499354148695951773219604919180708919708867187479, 1102507499354148695951773219604919180708919708867187480, 1102507499354148695951773219604919180708919708867187481, 1102507499354148695951773219604919180708919708867187482, 1102507499354148695951773219604919180708919708867187483, 1102507499354148695951773219604919180708919708867187484, 1102507499354148695951773219604919180708919708867187485, 1102507499354148695951773219604919180708919708867187486, 1102507499354148695951773219604919180708919708867187487, 1102507499354148695951773219604919180708919708867187488, 1102507499354148695951773219604919180708919708867187489, 1102507499354148695951773219604919180708919708867187490, 1102507499354148695951773219604919180708919708867187491, 1102507499354148695951773219604919180708919708867187492, 1102507499354148695951773219604919180708919708867187493, 1102507499354148695951773219604919180708919708867187494, 1102507499354148695951773219604919180708919708867187495, 1102507499354148695951773219604919180708919708867187496, 1102507499354148695951773219604919180708919708867187497, 1102507499354148695951785947156748908546646759033203125, 1102507499354148695951786035130848413273062653203124964, 1102507499354148695951786035130848413273062653203124965, 1102507499354148695951786035130848413273062653203124966, 1102507499354148695951786035130848413273062653203124967, 1102507499354148695951786035130848413273062653203124968, 1102507499354148695951786035130848413273062653203124969, 1102507499354148695951786035130848413273062653203124970, 1102507499354148695951786035130848413273062653203124971, 1102507499354148695951786035130848413273062653203124972, 1102507499354148695951786035130848413273062653203124973, 1102507499354148695951786035130848413273062653203124974, 1102507499354148695951786035130848413273062653203124975, 1102507499354148695951786035130848413273062653203124976, 1102507499354148695951786035130848413273062653203124977, 1102507499354148695951786035130848413273062653203124978, 1102507499354148695951786035130848413273062653203124979, 1102507499354148695951786035130848413273062653203124980, 1102507499354148695951786035130848413273062653203124981, 1102507499354148695951786035130848413273062653203124982, 1102507499354148695951786035130848413273062653203124983, 1102507499354148695951786035130848413273062653203124984, 1102507499354148695951786035130848413273062653203124985, 1102507499354148695951786035130848413273062653203124986, 1102507499354148695951786035130848413273062653203124987, 1102507499354148695951786035130848413273062653203124988, 1102507499354148695951786035130848413273062653203124989, 1102507499354148695951786035130848413273062653203124990, 1102507499354148695951786035130848413273062653203124991, 1102507499354148695951786035130848413273062653203124992, 1102507499354148695951786035130848413273062653203124993, 1102507499354148695951786035130848413273062653203124994, 1102507499354148695951786035130848413273062653203124995, 1102507499354148695951786035130848413273062653203124996, 1102507499354148695951786035130848413273062653203124997, 1102507499354148695951786420298270327866468231201171875, 1102507499354148695951786421355820618003786804199218714, 1102507499354148695951786421355820618003786804199218715, 1102507499354148695951786421355820618003786804199218716, 1102507499354148695951786421355820618003786804199218717, 1102507499354148695951786421355820618003786804199218718, 1102507499354148695951786421355820618003786804199218719, 1102507499354148695951786421355820618003786804199218720, 1102507499354148695951786421355820618003786804199218721, 1102507499354148695951786421355820618003786804199218722, 1102507499354148695951786421355820618003786804199218723, 1102507499354148695951786421355820618003786804199218724, 1102507499354148695951786421355820618003786804199218725, 1102507499354148695951786421355820618003786804199218726, 1102507499354148695951786421355820618003786804199218727, 1102507499354148695951786421355820618003786804199218728, 1102507499354148695951786421355820618003786804199218729, 1102507499354148695951786421355820618003786804199218730, 1102507499354148695951786421355820618003786804199218731, 1102507499354148695951786421355820618003786804199218732, 1102507499354148695951786421355820618003786804199218733, 1102507499354148695951786421355820618003786804199218734, 1102507499354148695951786421355820618003786804199218735, 1102507499354148695951786421355820618003786804199218736, 1102507499354148695951786421355820618003786804199218737, 1102507499354148695951786421355820618003786804199218738, 1102507499354148695951786421355820618003786804199218739, 1102507499354148695951786421355820618003786804199218740, 1102507499354148695951786421355820618003786804199218741, 1102507499354148695951786421355820618003786804199218742, 1102507499354148695951786421355820618003786804199218743, 1102507499354148695951786421355820618003786804199218744, 1102507499354148695951786421355820618003786804199218745, 1102507499354148695951786421355820618003786804199218746, 1102507499354148695951786421355820618003786804199218747, 1102507499354148695951786433413508348166942596435546875]),
(35, 34, 1, [1102507499354148695951786433413508348166942596435546875]),


 ]

"""#### 36-41 Counts only except for lists with len<10"""

table_all_equal_by_base += [
(36, 0, 1, [3831465379323568772890827210355149992132716389119437755185]),
(36, 1, 178549, None),
(36, 2, 16392, None),
(36, 3, 3984, None),
(36, 4, 1565, None),
(36, 5, 708, None),
(36, 6, 62, None),
(36, 7, 154, None),
(36, 8, 225, None),
(36, 9, 6, [22375196113925215127896927766653727337560989797187582, 27363003838404455969062107961022735015475271509737472, 1477602207273840622329353829895227690835664661525823488, 1499977403387765837457250757661881418173225651323011070, 1504965211112245078298415937856250425851139933035560960, 106387358923716524807713475752456393740167855629859291136]),
(36, 10, 112, None),
(36, 11, 77, None),
(36, 12, 2, [1477602207273840622329353829895227690835664661525823488, 106387358923716524807713475752456393740167855629859291136]),
(36, 13, 38, None),
(36, 14, 2, [1477602207273840622329353829895227690835664661525823488, 106387358923716524807713475752456393740167855629859291136]),
(36, 15, 2, [1477602207273840622329353829895227690835664661525823488, 106387358923716524807713475752456393740167855629859291136]),
(36, 16, 38, None),
(36, 17, 38, None),
(36, 18, 1, [106387358923716524807713475752456393740167855629859291136]),
(36, 19, 38, None),
(36, 20, 2, [104909756716442684185384121922561166049332190968333467648, 106387358923716524807713475752456393740167855629859291136]),
(36, 21, 2, [104909756716442684185384121922561166049332190968333467648, 106387358923716524807713475752456393740167855629859291136]),
(36, 22, 38, None),
(36, 23, 2, [104909756716442684185384121922561166049332190968333467648, 106387358923716524807713475752456393740167855629859291136]),
(36, 24, 3, [104909756716442684185384121922561166049332190968333467648, 105432148576616810623930244526107328467954487970039332862, 106387358923716524807713475752456393740167855629859291136]),
(36, 25, 112, None),
(36, 26, 7, [104882393712604279729415059814600143314316715696823730176, 104887381520328758970256224994794512321994629978536280062, 104909756716442684185384121922561166049332190968333467648, 105606301697105312639475564899062103496600638653930143742, 106359995919878120351744413644495371005152380358349553664, 106364983727602599592585578824689740012830294640062103550, 106387358923716524807713475752456393740167855629859291136]),
(36, 27, 77, None),
(36, 28, 154, None),
(36, 29, 358, None),
(36, 30, 412, None),
(36, 31, 1565, None),
(36, 32, 2208, None),
(36, 33, 11916, None),
(36, 34, 134499, None),
(36, 35, 1, [106387358923716524807713475752456393740167855629859291136]),
(37, 0, 0, []),
(37, 1, 30, None),
(37, 2, 25, None),
(37, 3, 24, None),
(37, 4, 25, None),
(37, 5, 29, None),
(37, 6, 9, [1141898494025569862863796716798677962490442463304734, 1141912520266233277047963691422646403848648648543644, 37837430499009591584731173514583310797053192626616442, 52945274654107916526610168038757323406214870871846945, 2605349383260997795578386962581716320914387715097632881, 2605349383260997795578386962601628495490963855152414744, 2605349383260997795578386962603027965435096866821756506, 143939633560505653584127699054686138733123330188365339561, 10555134955777783414078330085995832946127396083370199442517]),
(37, 7, 19, None),
(37, 8, 19, None),
(37, 9, 11, None),
(37, 10, 25, None),
(37, 11, 27, None),
(37, 12, 2, [143939633560505455873225396622692378773059507342033489216, 10555134955777783414078330085995832946127396083370199442517]),
(37, 13, 29, None),
(37, 14, 21, None),
(37, 15, 14, None),
(37, 16, 20, None),
(37, 17, 28, None),
(37, 18, 1, [10555134955777783414078330085995832946127396083370199442517]),
(37, 19, 29, None),
(37, 20, 21, None),
(37, 21, 15, None),
(37, 22, 23, None),
(37, 23, 28, None),
(37, 24, 2, [10411195322217277958205104689373140567354336576028165953301, 10555134955777783414078330085995832946127396083370199442517]),
(37, 25, 28, None),
(37, 26, 27, None),
(37, 27, 12, None),
(37, 28, 20, None),
(37, 29, 20, None),
(37, 30, 12, None),
(37, 31, 30, None),
(37, 32, 24, None),
(37, 33, 24, None),
(37, 34, 25, None),
(37, 35, 29, None),
(37, 36, 1, [10555134955777783414078330085995832946127396083370199442517]),
(38, 0, 0, []),
(38, 1, 4002, None),
(38, 2, 908, None),
(38, 3, 516, None),
(38, 4, 446, None),
(38, 5, 797, None),
(38, 6, 417, None),
(38, 7, 442, None),
(38, 8, 167, None),
(38, 9, 7, [191463703543733841713320782163451009756554135180171804670, 250049799229944995494369566055118832395399915737223004160, 7067406120871888211964961487448495181496981184221768843262, 14156734236578868185005648063896365514677465379347579797504, 14348197940122602026718968846059816524434019514527751602174, 14406784035808813180500017629951484347072865295084802801664, 1075911801979993982060429252856123779115487368830416064610304]),
(38, 10, 80, None),
(38, 11, 44, None),
(38, 12, 3, [9430708146072006986947741792196444583360800603064429334203, 14156734236578868185005648063896365514677465379347579797504, 1075911801979993982060429252856123779115487368830416064610304]),
(38, 13, 2, [14156734236578868185005648063896365514677465379347579797504, 1075911801979993982060429252856123779115487368830416064610304]),
(38, 14, 40, None),
(38, 15, 3, [11793432134006870386698603377894389086450905531976604712958, 14156734236578868185005648063896365514677465379347579797504, 1075911801979993982060429252856123779115487368830416064610304]),
(38, 16, 40, None),
(38, 17, 40, None),
(38, 18, 40, None),
(38, 19, 1, [1075911801979993982060429252856123779115487368830416064610304]),
(38, 20, 40, None),
(38, 21, 40, None),
(38, 22, 3, [1061755067743415113875423604792227413600809903451068484812800, 1064118369845987111673730649478229390029036463298439459897342, 1075911801979993982060429252856123779115487368830416064610304]),
(38, 23, 40, None),
(38, 24, 3, [1061755067743415113875423604792227413600809903451068484812800, 1065693331444928638668898465494375418870221941965754651901950, 1075911801979993982060429252856123779115487368830416064610304]),
(38, 25, 2, [1061755067743415113875423604792227413600809903451068484812800, 1075911801979993982060429252856123779115487368830416064610304]),
(38, 26, 44, None),
(38, 27, 80, None),
(38, 28, 7, [1061505017944185168879929235226172294768414503535331261808640, 1061563604039871380033710284010063962591053349315888313008126, 1061755067743415113875423604792227413600809903451068484812800, 1068844395859122093848464291368675283933990387646194295767038, 1075661752180764037064934883290068660283091968914678841606144, 1075720338276450248218715932073960328105730814695235892805630, 1075911801979993982060429252856123779115487368830416064610304]),
(38, 29, 162, None),
(38, 30, 442, None),
(38, 31, 417, None),
(38, 32, 797, None),
(38, 33, 446, None),
(38, 34, 516, None),
(38, 35, 908, None),
(38, 36, 3170, None),
(38, 37, 1, [1075911801979993982060429252856123779115487368830416064610304]),
(39, 0, 0, []),
(39, 1, 25162, None),
(39, 2, 2809, None),
(39, 3, 705, None),
(39, 4, 525, None),
(39, 5, 403, None),
(39, 6, 169, None),
(39, 7, 321, None),
(39, 8, 283, None),
(39, 9, 81, None),
(39, 10, 4, [20553029666019209137013969739827471119985905308004284775822, 24675684300256671606353155596951360931446878198377496605293, 1455865373715143624774836180220130294955365813704272299712287, 112595147462071192539789448988889059930192105219196517009951959]),
(39, 11, 81, None),
(39, 12, 43, None),
(39, 13, 3, [1013763857552043424405325943049910025226029340813084477063764, 1455865373715143624774836180220130294955365813704272299712287, 112595147462071192539789448988889059930192105219196517009951959]),
(39, 14, 42, None),
(39, 15, 3, [1431189689414886953168483024623178934023918935505894803106994, 1455865373715143624774836180220130294955365813704272299712287, 112595147462071192539789448988889059930192105219196517009951959]),
(39, 16, 5, [1248156156106726113217417724810996554071929649978811636012597, 1431189689414886953168483024623178934023918935505894803106994, 1437372214936735919749237383999386181916321825199467702293445, 1455865373715143624774836180220130294955365813704272299712287, 112595147462071192539789448988889059930192105219196517009951959]),
(39, 17, 4, [1431189689414886953168483024623178934023918935505894803106994, 1439433523163783304771699073403030383673736586348024517932546, 1455865373715143624774836180220130294955365813704272299712287, 112595147462071192539789448988889059930192105219196517009951959]),
(39, 18, 4, [1431189689414886953168483024623178934023918935505894803106994, 1441493394614842875940341560293654839363585200065892034905008, 1455865373715143624774836180220130294955365813704272299712287, 112595147462071192539789448988889059930192105219196517009951959]),
(39, 19, 1, [112595147462071192539789448988889059930192105219196517009951959]),
(39, 20, 4, [111139282088356048915014612808668929635236739405492244710239672, 111153654067456349663849107428595405090828520019130624975046947, 111163957772656305586620965964265880996168186283690622206844965, 112595147462071192539789448988889059930192105219196517009951959]),
(39, 21, 4, [111139282088356048915014612808668929635236739405492244710239672, 111155713938907409235017749915486029546518368632848492492019409, 111163957772656305586620965964265880996168186283690622206844965, 112595147462071192539789448988889059930192105219196517009951959]),
(39, 22, 5, [111139282088356048915014612808668929635236739405492244710239672, 111157775247134456620040211604889673748275783393997049307658510, 111163957772656305586620965964265880996168186283690622206844965, 111346991305964466426572031264078063376120175569217705373939358, 112595147462071192539789448988889059930192105219196517009951959]),
(39, 23, 4, [111139282088356048915014612808668929635236739405492244710239672, 111163957772656305586620965964265880996168186283690622206844965, 111425130274597145267686230784564409011861489123117255865023332, 112595147462071192539789448988889059930192105219196517009951959]),
(39, 24, 42, None),
(39, 25, 42, None),
(39, 26, 41, None),
(39, 27, 81, None),
(39, 28, 4, [111139282088356048915014612808668929635236739405492244710239672, 112570471777770935868183095833292108569260658340998139513346666, 112574594432405173330652435019149232459072119313888512725176133, 112595147462071192539789448988889059930192105219196517009951959]),
(39, 29, 81, None),
(39, 30, 283, None),
(39, 31, 321, None),
(39, 32, 169, None),
(39, 33, 403, None),
(39, 34, 529, None),
(39, 35, 693, None),
(39, 36, 2329, None),
(39, 37, 24001, None),
(39, 38, 1, [112595147462071192539789448988889059930192105219196517009951959]),
(40, 0, 0, []),
(40, 1, 103307, None),
(40, 2, 14138, None),
(40, 3, 3017, None),
(40, 4, 1134, None),
(40, 5, 46, None),
(40, 6, 296, None),
(40, 7, 257, None),
(40, 8, 87, None),
(40, 9, 166, None),
(40, 10, 46, None),
(40, 11, 167, None),
(40, 12, 8, [2450366474433685832183737588121599999999999999999999999999998, 2502854235920911963258880000000000000000000000000000000000000, 2550077900749608415395840000000000000000000000000000000000000, 151115727451828646838272000000000000000000000000000000000000000, 153566093926262332670455737588121599999999999999999999999999998, 153618581687749558801530880000000000000000000000000000000000000, 153665805352578255253667840000000000000000000000000000000000000, 12089258196146291747061760000000000000000000000000000000000000000]),
(40, 13, 2, [151115727451828646838272000000000000000000000000000000000000000, 12089258196146291747061760000000000000000000000000000000000000000]),
(40, 14, 3, [111295653384329123056166188892381485720745113276077442469783073, 151115727451828646838272000000000000000000000000000000000000000, 12089258196146291747061760000000000000000000000000000000000000000]),
(40, 15, 42, None),
(40, 16, 42, None),
(40, 17, 42, None),
(40, 18, 42, None),
(40, 19, 42, None),
(40, 20, 1, [12089258196146291747061760000000000000000000000000000000000000000]),
(40, 21, 42, None),
(40, 22, 42, None),
(40, 23, 42, None),
(40, 24, 42, None),
(40, 25, 2, [11938142468694463100223488000000000000000000000000000000000000000, 12089258196146291747061760000000000000000000000000000000000000000]),
(40, 26, 2, [11938142468694463100223488000000000000000000000000000000000000000, 12089258196146291747061760000000000000000000000000000000000000000]),
(40, 27, 8, [11935592390793713491808092160000000000000000000000000000000000000, 11935639614458542188260229120000000000000000000000000000000000000, 11935692102220029414391304262411878399999999999999999999999999998, 11938142468694463100223488000000000000000000000000000000000000000, 12086708118245542138646364160000000000000000000000000000000000000, 12086755341910370835098501120000000000000000000000000000000000000, 12086807829671858061229576262411878399999999999999999999999999998, 12089258196146291747061760000000000000000000000000000000000000000]),
(40, 28, 167, None),
(40, 29, 128, None),
(40, 30, 84, None),
(40, 31, 89, None),
(40, 32, 128, None),
(40, 33, 296, None),
(40, 34, 24, None),
(40, 35, 642, None),
(40, 36, 1049, None),
(40, 37, 14138, None),
(40, 38, 74295, None),
(40, 39, 1, [12089258196146291747061760000000000000000000000000000000000000000]),
(41, 0, 0, []),
(41, 1, 31, None),
(41, 2, 27, None),
(41, 3, 33, None),
(41, 4, 18, None),
(41, 5, 14, None),
(41, 6, 17, None),
(41, 7, 33, None),
(41, 8, 9, [4173178060473984186866797717287776342402677485142325908051070, 4899363873511971374463255709880795896255762266476002368962924, 265539042992896582459295605114515015137935809945436099456347989, 265539042992896582459295605114519991725625538662064150030886376, 265539042992896582459295605114523232154175794488373791861055069, 265539060407677206196338044900190428107312996745416273747538919, 265539083142572775731192157780095923162906972060862167120846578, 16365434231224249631027485273553574739039983126711856008803301551, 1330877630632711998713399240963346255985889330161650994325137953641]),
(41, 9, 33, None),
(41, 10, 6, [265539042992896582459295604463421319225415303373343374251110082, 265539076976177411644268930207658185314980270236393399475758499, 265539083142572775731192157780095731181987302479289686795585695, 8312775202963886610363044458430388234333581801223850545763414405, 16365434231224249631025131454937142335477309457974366438131362759, 1330877630632711998713399240963346255985889330161650994325137953641]),
(41, 11, 29, None),
(41, 12, 22, None),
(41, 13, 19, None),
(41, 14, 31, None),
(41, 15, 22, None),
(41, 16, 19, None),
(41, 17, 23, None),
(41, 18, 30, None),
(41, 19, 30, None),
(41, 20, 1, [1330877630632711998713399240963346255985889330161650994325137953641]),
(41, 21, 28, None),
(41, 22, 29, None),
(41, 23, 25, None),
(41, 24, 20, None),
(41, 25, 21, None),
(41, 26, 31, None),
(41, 27, 17, None),
(41, 28, 23, None),
(41, 29, 28, None),
(41, 30, 6, [1314512196401487749082374109508409113650412020703676627887006590882, 1322564855429748112103036196504915867751555748360427143779374539232, 1330612091549569425937668048805566160254707342859171704638342367946, 1330612091555735821301754972033138597800574349891414600925662195138, 1330612091589719102130939945358882834666663914858277650950886843559, 1330877630632711998713399240963346255985889330161650994325137953641]),
(41, 31, 33, None),
(41, 32, 9, [1314512196401487749082371755689792681246849347034939138316334652090, 1330612091549569425937668048805566160062726423189590132158017107063, 1330612091572304321507202902918446065557782017164905578051390414719, 1330612091589719102130939945358231732753735154367162620533276898572, 1330612091589719102130939945358231735994163704622988930175107067261, 1330612091589719102130939945358231740970751394351705558225681605652, 1330872731268838486742024777707636375189993074399384518322768990717, 1330873457454651524729212374165628968209546927484165851999229902567, 1330877630632711998713399240963346255985889330161650994325137953641]),
(41, 33, 33, None),
(41, 34, 18, None),
(41, 35, 14, None),
(41, 36, 19, None),
(41, 37, 33, None),
(41, 38, 27, None),
(41, 39, 34, None),
(41, 40, 1, [1330877630632711998713399240963346255985889330161650994325137953641]),
]

"""### Extending https://oeis.org/A165617"""

print([lgth for base, digit, lgth, seq in table_all_equal_by_base if digit == 1])

"""### 10 in various bases"""

if not "SKIP":
  # Populate table_all_equal_by_base below
  for base in range(19, 101):
    mues, mugs = multidigit_unbounded_eq_search(base, base=base, sizebound=1100)
    print(f"({base}, {mues}, {mugs}),")

ten_in_each_base = [
(2, 21, 22),
(3, 8733, 8865),
(4, 335544292, 339640321),
(5, 2861022949218705, 2894950312181761),
(6, 9285104628270801031790526, 9324292176657900371968001),
(7, 17470515007934372792874498022030658653, 18520243546832041183519349807125626881),
(8, 26938398179283202588330318836626528618298148632133512, 27027567899607851881890043883659636609971625863938049),
(9, 45677590745077404064777874376752672121786802517249749853726469790339137, 46510990767946274970757935935441037553026400050102960914879080325709825),
(10, 109999999999999999999999999999999999999999999999999999999999999999999999999999999999999999810, 110363899000137033701953258110096424585284286687247951331239666673017274235191757537856716801),
(11, 471812343446070425886410080642593080457139291969630234260416142681041845599523778243767795467124336667938437482119101, 476436028497917456605404326337373749657227590905530106609343680909013443738940143685287541444653571728417002356736001),
(12, 4416203490206766347244363875316682483125460561898677483681522677550253813853872070638938141513829984197431742609896385776433498928087198897012460, 4453700573603478330692807849741994660856245980966042152286143176006003079863734021824669625876395050527678807854432651092699911885324600710529025),
(13, 108451196518065273326866211246708463582660775568005800811678242885830666513712007346401488353870949948248796826851027537604630014878835800365170739743265780183374184434077649937, 111252770309564390809436992771006310380367594635927470057070779126105767310171054686257928881711361964939000040221868635756659090348860949818804095734083835912916382418764562433),
(14, 8270309989469883363902996967941057741652999666609810493466546688142753273170587413024102501785864415550020212469378287810393531004696099792530453453887485053370348921658927670727025117474561232235770593578843782, 8481824011716313559108999164752389903764155767907104271678351676345610430406665585818841385954443282246566741424194221408133366713274019988444217928612704289903754738160348256098862743089926305473432391660863489),
(15, 2287576589346864749359729620629792024908216920672304560646843832864992722322903280079758872317025789805413088765584879846584098298015563958159674561964329803396554737635118967279428583330197707356809007491968421810213385469978675246238708496093749565, 2386594074239287451821826975364886518843321645244755758703805204813783468862820321697377262598104048336123808920232784929277203011958088377976668432516709122422310582357115741604442903331568671988091625542445783631879950513125146815172830656993427457),
(16, 2650725411100799749856120009502051220558333705265842047308070556361922043888467899870647297919906790509282937214943157745900894203335241580669212544597041730112507534388160664124854786293294592351961489426347399053642482635707405985243010903407558646689144954444363492806257376477800638184976, 2652974877229803988517679696327762714529317235236590401880778550354352407914930352438610555031256858318158757275280256280337437062933938612061267004709326122070331328723434711074595731304946329459580436630685997939355079166822704589418519601306523122488648636027350317055219167750542021623809),
(17, 14717296566164988322206641860850693134717595458442256458909891949544532058150821596500514130233353999916870384095126830239896456767908205436930307603780282324181571219953546727351213621901177708427184954201061208517982647833173503918852242954976681383064907309236499379022656589829001746127340985600859157771107636539909524725437493199105, 15109782130374556395702646977199542289571614059073839887670224498880149025009740498042818859949590580166173457092047783812199925686499075731553234466713519737878042595514939106261721971529225071437575822107150731482713488899650112708467290335766338877117153618481283499897815010415033899420832968107246111861109045567402154760521069363201),
(18, 444030362265687084629029022588732792071869600474457271569128571376738156444225395307610389963988072095502020529291038029361441310794265119061182943197880619868025904259449682874101243572437859431322201064894704519671356005707378825812198479538817565932359094169595502918076222903814959234134180221470311732024970596149963187016152780415883076517816699030927925851265356413569468473539978, 461827839598716657120741899481438413605677731836228406400517456059602358184216664610426903303116397631479702285250534144492762657769488658199316369309818514882566077384563051516678626351446492236987726252113460797303453380929837670554621478448517586342451336460863435621688617746717225272688253117938016289143672214594877383808085051753950626240459827188356569332842787528143885161201665),
(19, 81943615611170727302338645730680234902128641414800624409776294285728508828138161677966815996830854324684266531784322102549701177103439160905109082692466353110515992599035552377279945470906985638827030928638391929395348456925498460325352080334420981511593678875099124785466141690454545004256301634224235133151501621235515378418529844584398521837564815523106473339698595150083567714456474241805465318105014389557553384886633286813032073472477, 82074425320741644448825548650830265947684778171011736034032651772449769886945535739796580134898926698407886769693588593899831532061095187095035188584235076257098140094206149763282364883126379105181904486482465827126259453673298890393007272860159494948931820244041575550285753397688918560244253670607837825185667744735227007179026421289864525206887904964229585224146115157319548426692835287781869711285383678232114530925423658229114383368193),

]

[(b, len(strbase(eq, b)), letterbase(eq, b), letterbase(gt, b)) for b, eq, gt in ten_in_each_base]

"""Hey, if you look at the second column of that (or the third, it's the same), you get https://oeis.org/A027688 = n^2 + n + 3. Yeah! The base-10 version isn't in the OEIS, but it's also a lot less meaningful.

### Powers of 2 in base 2
"""

if "SKIP":
  # Populate powers_of_2_in_base_2
  for power in range(10, 11):
    eqs, n, chk = multidigit_all_eq_search(2**power, base=2, sizebound=1200)
    print(f"({power}, {n}, {eqs}),")

def consecutive(start, count):
  return list(range(start, start+count))

powers_of_2_in_base_2 = [
    (0, 1, [1]),
    (1, 1, [21]),
    (2, 1, [610]),
    (3, 1, [283187]),
    (4, 1, [35609822115]),
    (5, 1, [300185978028231432373]),
    (6, 1, [10978492537426792762986527405022424940922]),
    (7, 0, []), # Checked up to 1200 digits
    (8, 1, [1719605233816430810082212871358724351336084193513828012947053856850589611682582063581859138633460792569575136101572603133684419606722080444292650722578661374]),
    (9, 1, [46066288024135617632597874326280487639985658662885492290549686629676676354285339585007650804271491312641006292072876334746819118237476619820729754289719476147564561870287404002890126747713055107876038412285431484943014761787582590828689812640163747676909901958704492988503885235684765532336774786906520237758770]),
    (10, 1023, consecutive(16554426168460005920227414226732347463191622579033084941083314352956469696915158854597499541097741866953042964291158425834366731873423024245894595302680893963456004814351544891930709655284076703776686744660915399141854107465834153547098552509367136874873636858808670651601058184138772981251050130474844668063149793844982190028540248236661581198379102755600943936425343183821089770391450336963315243111101008005788583819055812585766209833455127225501614823241911827115975633626327796080216035243156976011370985817309151896886019953059692661025406018503434309413054100231715458805570577928233256155039505062468929869314049, 1023)),
]

for power, n, eqs in powers_of_2_in_base_2:
  if eqs:
    print(f"{power} {len(str(eqs[0]))} {len(strbase(eqs[0], 2))}_2")

"""### Non-decreasing portions of the eq sequence"""



nondecbase = defaultdict(int)
by_digit = defaultdict(lambda: list((0,)))
nondecdigit = defaultdict(int)
prev = 0
for base, digit, eq, gt in table_bases[1:]:
  if digit == 0:
    prev = 0
  else:
    if eq >= prev:
      nondecbase[base] += 1
    prev = eq
  if digit > 1:
    by_digit[digit].append(eq)
    if by_digit[digit][-2] <= by_digit[digit][-1]: nondecdigit[digit] += 1
nondecbase, nondecdigit

eqlater = defaultdict(int)
prev = 0
for base, digit, eq, gt in table_bases[1:]:
  if eq and eq > gt:
    eqlater[base] += 1
eqlater

for base, neqlate in eqlater.items():
  print(f"{base}\t{neqlate}\t{neqlate / base}")

uniqeq = defaultdict(set)
for base, digit, eq, gt in table_bases[1:]:
  uniqeq[base].add(eq)
for base, uset in uniqeq.items():
  print(f"{base}\t{len(uset)}\t{len(uset) / base}")

"""## Decilliness

Another "natural" way to number tapes (solely natural for a peculiar kind of mathematician) is to use the letters linguistically rather than in all their combinations, by labeling the first tape as "ONE", the second as "TWO", and so on. In this case, the sequences should be parameterized by a letter sticker that they use -- these sticker sheets did have letters on them -- and the language in which you wish to spell out these numbers.

Some values are clearly out, e.g. a_>("j", EN), also k, p, must be undefined because no spelled out numbers contain "x"s. Some will be very small: a_=("o", EN)=2 ("One", "twO"), and a_=("e", EN)=3 ("onE", "two", "thrEE"). Some will be very large indeed, because e.g. "a" first appears at "one thousand", and "c" famously does not appear until "one octillion". One can measure this with either cardinal ("one") or ordinal ("first") numbers, so perhaps that is another parameter, but for simplicity here we will use cardinals.

Unfortunately the most convenient package for this tops out at decillions, so there may exist members of the a_ sequences that we simply cannot find without doing more language work.
"""

import inflect
p = inflect.engine()
assert "one thousand, two hundred and thirty-four" == p.number_to_words(1234)
assert "one thousand, two hundred and thirty-fourth" == p.number_to_words(p.ordinal(1234))
assert ("one decillion, two nonillion, three octillion, four septillion, five sextillion, six quintillion, seven quadrillion, eight trillion, nine billion, ten million, eleven thousand and twelve"
        == p.number_to_words(1002003004005006007008009010011012))
# p.number_to_words(3131879606492502363897623259852309766) #  raises NumOutOfRangeError. We max out at decillions.
assert "three hundred and thirteen decillion, one hundred and eighty-seven nonillion, nine hundred and sixty octillion, six hundred and forty-nine septillion, two hundred and fifty sextillion, two hundred and thirty-six quintillion, three hundred and eighty-nine quadrillion, seven hundred and sixty-two trillion, three hundred and twenty-five billion, nine hundred and eighty-five million, two hundred and thirty thousand, nine hundred and seventy-six" == p.number_to_words(313187960649250236389762325985230976)

def slow_count_letters(limit):
  counts = defaultdict(int)
  eqs = {}
  gts = {}
  for i in tqdm(range(1, limit+1)):
    for char in p.number_to_words(i):
      counts[char] += 1
    for char, count in counts.items():
      if i > 1 and char not in eqs and count == i:
        eqs[char] = i
      if i > 1 and char not in gts and count > i:
        gts[char] = i
  return counts, eqs, gts

one, _, _ = slow_count_letters(1)
assert one["o"] == 1, one
assert one["n"] == 1, one
assert one["e"] == 1, one
two, _, _ = slow_count_letters(2)
assert two["o"] == 2, two
assert two["n"] == 1, two
assert two["e"] == 1, two
assert two["t"] == 1, two
assert two["w"] == 1, two
three, _, _ = slow_count_letters(3)
assert three["o"] == 2, three
assert three["n"] == 1, three
assert three["e"] == 3, three
assert three["t"] == 2, three
assert three["w"] == 1, three
assert three["h"] == 1, three
assert three["r"] == 1, three

decilliness_9x5 = ({'o': 205000,
                    'n': 503901,
                    'e': 462000,
                    't': 403000,
                    'w': 50000,
                    'h': 289000,
                    'r': 190000,
                    'f': 122000,
                    'u': 219000,
                    'i': 222000,
                    'v': 82000,
                    's': 199000,
                    'x': 50000,
                    'g': 50000,
                    'l': 4000,
                    'y': 160000,
                    '-': 144000,
                    ' ': 475902,
                    'd': 377901,
                    'a': 197901,
                    ',': 89100},
                  {'o': 2,
                    'e': 3,
                    't': 23,
                    'n': 30,
                    'd': 149,
                    'r': 240,
                    'h': 243,
                    'u': 453,
                    'a': 1109,
                    'f': 5556,
                    'y': 24998,
                    '-': 29172},
                  {'e': 11,
                    't': 24,
                    'n': 29,
                    ' ': 150,
                    'd': 150,
                    'r': 241,
                    'h': 248,
                    'o': 251,
                    'u': 454,
                    'i': 559,
                    'a': 1110,
                    's': 1716,
                    'f': 5557,
                    'y': 24999,
                    '-': 29173})

decilliness_9x7 = (
             {'o': 35590000,
              'n': 85399901,
              'e': 71200000,
              't': 46390000,
              'w': 7000000,
              'h': 41990000,
              'r': 32000000,
              'f': 16200000,
              'u': 32990000,
              'i': 48200000,
              'v': 12200000,
              's': 23990000,
              'x': 7000000,
              'g': 7000000,
              'l': 18400000,
              'y': 16000000,
              '-': 14400000,
              ' ': 92599902,
              'd': 64799901,
              'a': 28799901,
              ',': 17990100,
              'm': 9000000},
 {'o': 2,
  'e': 3,
  't': 23,
  'n': 30,
  'd': 149,
  'r': 240,
  'h': 243,
  'u': 453,
  'a': 1109,
  'f': 5556,
  'y': 24998,
  '-': 29172,
  'v': 1104051,
  ',': 1113298,
  'l': 1921566},
 {'e': 11,
  't': 24,
  'n': 29,
  ' ': 150,
  'd': 150,
  'r': 241,
  'h': 248,
  'o': 251,
  'u': 454,
  'i': 559,
  'a': 1110,
  's': 1716,
  'f': 5557,
  'y': 24999,
  '-': 29173,
  'v': 735759,
  ',': 1113299,
  'l': 1921567})

if  "SKIP":
  #slow_count_letters(int("9" * 36))
  slow_count_letters(int("9" * 5))

"""## Extras

We played with the idea of skipping to the next number not containing the digit of interest, but in the end we only used it conceptually, not in any applied setting.
"""

for expected, source, base in [(0, [], 2),
                               (1, [1], 2),
                               (2, [1, 0], 2),
                               (3, [1, 1], 2),
                               (2, [2], 2),
                               (4, [1, 0, 0], 2),
                               (4, [2, 0], 2),
                               (5, [1, 0, 1], 2),
                               (6, [1, 0, 2], 2),
                               (0, [0], 10),
                               (11, [1, 1], 10),
                               (10, [1, 0], 10),
                               (11, [0, 11], 10),
                               (110, [11, 0], 10),
                               (234, [2, 3, 4], 10),
                               (1000, [9, 9, 10], 10),
                               ]:
  observed = unstrbase(source, base)
  assert expected == observed, f"u({source}, {base}) == {expected} but == {observed}"

def next_noncontaining(digit, limit, base=10, debug=False):
  digits = strbase(limit, base)
  if base == 2 and digit == 1:
    return None
  if digit in digits:
    result = []
    changed = False
    if digit == 0:
      for d in digits:
        if d == 0:
          changed = True
        if changed:
          result.append(1)
        else:
          result.append(d)
    else:
      for d in digits:
        if changed:
          result.append(0)
        elif d == digit:
          changed=True
          result.append(d + 1)
        else:
          result.append(d)
    return unstrbase(result, base)
  else:
    return limit

for b, d, n, expected in [(10, 0, 1, 1),
                          (10, 0, 10, 11),
                          (10, 0, 100, 111),
                          (10, 0, 303, 311),
                          (10, 0, 90907, 91111),
                          (10, 9, 90909, 100000),
                          (10, 9, 10009, 10010),
                          (10, 9, 12349, 12350),
                          (10, 9, 12934, 13000),
                          (2, 0, 0, 1),
                          (2, 0, 8, 15), # 1000 -> 1111 = 15
                          (2, 1, 8, None),  # There are no binary numbers without a 1 in them.
                          (6, 0, 37, 43),  # 101 -> 111 = 43
                          (6, 1, 37, 72),  # 101 -> 200 = 72
                          ]:
  nnc = next_noncontaining(d, n, base=b)
  assert expected == nnc, f"{b=} {d=} {n=} {expected=} {nnc=}"