A053778 First of four consecutive primes that comprise two sets of twin primes.

5, 11, 101, 137, 179, 191, 419, 809, 821, 1019, 1049, 1481, 1871, 1931, 2081, 2111, 2969, 3251, 3359, 3371, 3461, 4217, 4229, 4259, 5009, 5651, 5867, 6689, 6761, 6779, 6947, 7331, 7547, 8219, 8969, 9419, 9431, 9437, 10007, 11057, 11159, 11699, 12239

Offset: 1

Author

Labos Elemer, Mar 24 2000

Keywords

• nonn

Comments

These twins are not necessarily at the minimal distance as in A007530 (which is a subsequence).

Examples

			These primes initiate consecutive p quadruples as follows: [p,p+2,p+6k,p+6k+2]. For 6k=6,12,18,24,30,36,54 such a p =5,137,1931,9437,2968, 20441 and 48677 resp. Such a quadruple is [48677,48679,48731,48733], with [2,52,2] difference pattern.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..5274.
• Eric Weisstein's World of Mathematics, Prime Quadruplet

Crossrefs

Cf. A001223, A001359, A007530, A048614.

Programs

Mathematica

Transpose[Select[Partition[Prime[Range[1500]],4,1],#[[4]]-#[[3]]== #[[2]]-#[[1]]== 2&]][[1]] (* Harvey P. Dale, Jul 07 2011 *)

PARI

forprime( p=1,10^5, isprime(p+2) || next; isprime(nextprime(p+4)+2) && print1(p","))

PARI

nextA053778(p)=until( isprime(nextprime(p+1)+2), until( p+2==p=nextprime(p+1),)); p-2

PARI

p=0; A053778=vector(100,i, p=nextA053778(p+1))

Formula

A001359 primes for which A048614 is zero. Lesser of 2-twin primes after which the consecutive prime difference pattern (of A001223) is [2, 6k-2, 2] for some k.

Extensions

Edited by N. J. A. Sloane, Apr 13 2008, at the suggestion of M. F. Hasler.