A022007 Initial members of prime 5-tuples (p, p+4, p+6, p+10, p+12).

7, 97, 1867, 3457, 5647, 15727, 16057, 19417, 43777, 79687, 88807, 101107, 257857, 266677, 276037, 284737, 340927, 354247, 375247, 402757, 419047, 427237, 463447, 470077, 626617, 666427, 736357, 823717, 855727, 959467, 978067, 1022377, 1043587, 1068247

Offset: 1

Author

Warut Roonguthai

Keywords

• nonn
• easy

Comments

Subsequence of A052378. - R. J. Mathar, Feb 11 2013

All terms are congruent to 7 (modulo 30). - Matt C. Anderson, May 22 2015

This sequence is related to the admissible prime 5-tuple (0, 4, 6, 10, 12) because the sequence [1, 2, 3, 1, 2, repeat(1)] gives for n >= 1 the smallest element of RS0(A000040(n)) (the smallest nonnegative complete residue systems modulo prime(n)) which defines a residue class containing none of the 5-tuple members. This 5-tuple is one of two prime constellations of diameter 12. The other one is (0, 2, 6, 8, 12) with initial members given in A022006. See the Wikipedia and Weisstein pages. - Wolfdieter Lang, Oct 06 2017

Examples

			Admissibility guaranteeing sequence [1, 2, 3, 1, 2, repeat(1)] examples: the only residue class modulo prime(3) = 5 which contains none of the 5-tuple (0, 4, 6, 10, 12) members is 3 (mod 5). For prime(5) = 11 the first class is 2 (mod 11); the others are 3, 5, 7, 8, 9 (mod 11). - _Wolfdieter Lang_, Oct 06 2017
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• T. Forbes and Norman Luhn, Prime k-tuplets.
• Norman Luhn, Table of n, a(n) for n = 1..1000000.
• Eric Weisstein's World of Mathematics, Prime Constellation.
• Wikipedia, Prime k-tuple.

Crossrefs

Cf. A022006, A089157.

Programs

Magma

[p: p in PrimesUpTo(2*10^6) | IsPrime(p+4) and IsPrime(p+6) and IsPrime(p+10)and IsPrime(p+12)]; // Vincenzo Librandi, Aug 23 2015

Mathematica

Transpose[Select[Partition[Prime[Range[76000]], 5, 1], Differences[#] == {4, 2, 4, 2} &]][[1]] (* Harvey P. Dale, Aug 16 2014 *)

PARI

forprime(p=2,1e5,if(isprime(p+4)&&isprime(p+6)&&isprime(p+10)&&isprime(p+12),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

Perl

use ntheory ":all"; say for sieve_prime_cluster(1,1e7, 4,6,10,12); # Dana Jacobsen, Sep 30 2015

Formula

a(n) = 7 + 30*A089157(n). - Zak Seidov, Nov 01 2011