%I A400361 #17 Oct 08 2026 17:24:59
%S A400361 0,0,0,0,0,1,0,0,2,2,0,5,0,2,8,4,0,7,0,12,14,5,0,16,12,4,12,19,0,32,0,
%T A400361 13,22,6,34,38,0,7,24,44,0,49,0,28,60,11,0,52,30,40,34,37,0,52,62,75,
%U A400361 40,12,0,109,0,10,90,48,68,81,0,50,48,115,0,128,0
%N A400361 a(n) = A018804(n) - A384628(n).
%C A400361 a(n) is from [0, A018804(n) - A000203(n) - n + A000005(n)].
%C A400361 The analogous version a(n) = Sum_{k=1..n} (gcd(n,k) - gcd(n, ceiling(n/k))) does not stay nonnegative (example n = 6).
%C A400361 Sum_{k=1..n} a(k) = (3/Pi^2)*n^2*log(n) + O(n^2).
%C A400361 Conjecture: Sum_{k=1..n} a(k) = (3/Pi^2)*n^2*log(n) + C*n^2 + o(n^2), with C = (3/Pi^2)*(-1/2 + 2*gamma - 6*zeta'(2)/Pi^2) - Pi^2/12 - (1/2)*Sum_{d>=1} phi(d)*(digamma(1+1/d) + gamma)/d^2 = -1.2036....
%F A400361 a(n) = 0 for n from {A000040} U {1,4,8}.
%F A400361 a(n) = A018804(n) - A000203(n) - Sum_{k=1..n, k does not divide n} gcd(n, floor(n/k)).
%e A400361 For n = 1: a(1) = A018804(1) - A384628(1) = 0.
%t A400361 f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := Times @@ f @@@ FactorInteger[n] - Sum[GCD[n, Floor[n/k]], {k, 1, n}]; Array[a, 100] (* _Amiram Eldar_, Sep 24 2026 *)
%o A400361 (Python)
%o A400361 from math import prod, gcd
%o A400361 from sympy import factorint
%o A400361 def A400361(n):
%o A400361     c, j = prod(p**(e-1)*((p-1)*e+p) for p, e in factorint(n).items()), 1
%o A400361     while j <= n:
%o A400361         k = n//j
%o A400361         m = n//k
%o A400361         c -= (m-j+1)*gcd(n,k)
%o A400361         j = m+1
%o A400361     return c # _Chai Wah Wu_, Oct 02 2026
%Y A400361 Cf. A000005, A000040, A000203, A018804, A272718, A384628, A385398, A385402, A385858.
%K A400361 nonn,easy,new
%O A400361 1,9
%A A400361 _Ctibor O. Zizka_, Sep 24 2026