%I A007530 M3816 #127 Jul 31 2026 02:35:15
%S A007530 5,11,101,191,821,1481,1871,2081,3251,3461,5651,9431,13001,15641,
%T A007530 15731,16061,18041,18911,19421,21011,22271,25301,31721,34841,43781,
%U A007530 51341,55331,62981,67211,69491,72221,77261,79691,81041,82721,88811,97841,99131
%N A007530 Prime quadruples: numbers k such that k, k+2, k+6, k+8 are all prime.
%C A007530 Except for the first term, 5, all terms == 11 (mod 30). - _Zak Seidov_, Dec 04 2008
%C A007530 Some further values: For k = 1, ..., 10, a(k*10^3) = 11721791, 31210841, 54112601, 78984791, 106583831, 136466501, 165939791, 196512551, 230794301, 265201421. - _M. F. Hasler_, May 04 2009
%C A007530 k is the first prime of 2 consecutive twin prime pairs. - _Daniel Forgues_, Aug 01 2009
%C A007530 The prime quadruples of form p + (0, 2, 6, 8) have the quadruple congruence class (-1, +1, -1, +1) (mod 6). - _Daniel Forgues_, Aug 12 2009
%C A007530 s = (p+8)-(p) = 8 is the smallest s giving an admissible prime quadruple form, for which the only admissible form is p + (0, 2, 6, 8), since (0, 2, 6, 8) is the only form not covering all the congruence classes for any prime <= 4. Since s is smallest, these prime quadruples are prime constellations (or prime quadruplets), i.e., they contain consecutive primes. - _Daniel Forgues_, Aug 12 2009
%C A007530 Except for the first term, 5, all prime quadruples are of the form (15k-4, 15k-2, 15k+2, 15k+4), with k >= 1, and so are centered on 15k. - _Daniel Forgues_, Aug 12 2009
%C A007530 Subsequence of A022004. - _R. J. Mathar_, Feb 10 2013
%C A007530 The quadruplets are listed in A136162. - _M. F. Hasler_, Apr 20 2013
%C A007530 Starting at a(2) and examining the first 50 terms, (a(n)+4)/15 is a prime in 8 cases and a semiprime in 21; the last 18 terms have 2 primes and 11 semiprimes. Do the number of semiprimes continue to occur greater than mere chance? - _J. M. Bergot_, Apr 27 2015
%C A007530 The values of (a(n)+4)/15 are listed in A112540, and all such values are odd. The value of (a(51)+4)/15 = 12991. There are 2527 odd semiprimes <= 12991 and 1547 primes <= 12991. So ~0.39 of all odd numbers in this range are semiprimes, and ~0.24 of all odd numbers in this range are primes. Thus, the occurrence of 21 semiprimes among 50 terms deviates very little from the amount expected by mere chance, and the occurrence of 8 primes among 50 terms is somewhat fewer than the amount expected by mere chance. - _Geoffrey Caveney_, Jul 30 2026
%D A007530 H. Rademacher, Lectures on Elementary Number Theory. Blaisdell, NY, 1964, p. 4.
%D A007530 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007530 Matt C. Anderson, <a href="/A007530/b007530.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe).
%H A007530 C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=Quadruple">prime quadruple</a>
%H A007530 Tony Forbes and Norman Luhn, <a href="http://www.pzktupel.de/ktuplets">prime k-tuplets</a>
%H A007530 Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
%H A007530 Norman Luhn, <a href="http://www.pzktupel.de/smarchive.html">Table of n, a(n) for n = 1..1000000</a>
%H A007530 Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/quads/quads.html">Enumeration to 1.6e15 of the prime quadruplets</a>
%H A007530 H. Riesel, <a href="https://doi.org/10.1007/978-0-8176-8298-9">Prime numbers and computer methods for factorization</a>, Progress in Mathematics, Vol. 57, Birkhäuser, Boston, 1985, ISBN: 978-0-8176-8297-2, Chap. 4, see p. 65.
%H A007530 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeQuadruplet.html">Prime Quadruplet</a>
%F A007530 a(n) = 11 + 30*A014561(n-1) for n > 1. - _M. F. Hasler_, May 04 2009
%e A007530 From _M. F. Hasler_, May 04 2009: (Start)
%e A007530 a(1)=5 is the start of the first prime quadruplet, {5,7,11,13}.
%e A007530 a(2)=11 is the start of the second prime quadruplet, {11,13,17,19}, and all other prime quadruplets differ from this one by a multiple of 30.
%e A007530 a(100)=470081 is the start of the 100th prime quadruplet;
%e A007530 a(500)=4370081 is the start of the 500th prime quadruplet.
%e A007530 a(167)=1002341 is the least quadruplet prime beyond 10^6. (End)
%t A007530 A007530 = Select[Range[1, 10^5 - 1, 2], Union[PrimeQ[# + {0, 2, 6, 8}]] == {True} &] (* _Alonso del Arte_, Sep 24 2011 *)
%t A007530 (* Alternative: *)
%t A007530 Select[Prime[Range[10000]],AllTrue[#+{2,6,8},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Mar 11 2019 *)
%o A007530 (PARI) A007530( n, print_all=0, s=2 )={ my(p,q,r); until(!n--, until( p+8==s=nextprime(s+2), p=q; q=r; r=s); print_all && print1(p","));p} \\ The optional 3rd argument can be used to obtain large values by starting from some precomputed point instead of zero, using a(n+k) = A007530(k+1,,a(n)) (or A007530(k,,a(n)-1) for k>0); e.g., you get a(10^4+k) using A007530(k+1,,265201421) (value of a(10^4) from the comments section). - _M. F. Hasler_, May 04 2009
%o A007530 (PARI) forprime(p=2, 10^5, if(isprime(p+2) && isprime(p+6) && isprime(p+8), print1(p, ", "))) \\ _Felix Fröhlich_, Jun 22 2014
%o A007530 (Magma) [ p: p in PrimesUpTo(11000)| IsPrime(p+2) and IsPrime(p+6) and IsPrime(p+8)]; // _Vincenzo Librandi_, Nov 18 2010
%o A007530 (Python)
%o A007530 from sympy import primerange
%o A007530 def aupto(limit):
%o A007530   p, q, r, alst = 2, 3, 5, []
%o A007530   for s in primerange(7, limit+9):
%o A007530     if p+2 == q and p+6 == r and p+8 == s: alst.append(p)
%o A007530     p, q, r = q, r, s
%o A007530   return alst
%o A007530 print(aupto(10**5)) # _Michael S. Branicky_, May 11 2021
%Y A007530 Cf. A159910 (first differences divided by 30), A120120, A007811, A014561.
%K A007530 nonn
%O A007530 1,1
%A A007530 _N. J. A. Sloane_, _Robert G. Wilson v_
%E A007530 More terms from _Warut Roonguthai_
%E A007530 Incorrect formula and Mathematica program removed by _N. J. A. Sloane_, Dec 04 2008, at the suggestion of _Zak Seidov_
%E A007530 Values up to a(1000) checked with the given PARI code by _M. F. Hasler_, May 04 2009