A105414 Primes p = prime(k) such that p+2 and prime(k+7)-2 are both prime numbers.

17, 71, 149, 191, 431, 521, 821, 1049, 1277, 1289, 1451, 1619, 1667, 1877, 1949, 2027, 2657, 3299, 3329, 3467, 3527, 3539, 3767, 3929, 4271, 4931, 5477, 5849, 6131, 6659, 6701, 6779, 6827, 8537, 8819, 8999, 9419, 9719, 9929, 10037, 10091, 11069, 11117

Offset: 1

Author

Cino Hilliard, May 02 2005

Keywords

• nonn

Comments

Conjecture: There are infinitely many primes p(k) such that p(k)-2 and p(k+m)-2 are both primes for all m > 1.

Examples

			p(8)-2 = 17, p(8+6)-2 = 41, both prime, 17 is in the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Crossrefs

Cf. A105409, A105410, A105411, A105412, A105413.

Programs

Mathematica

For[n = 1, n < 500, n++, If[PrimeQ[Prime[n] + 2], If[PrimeQ[Prime[n + 7] - 2], Print[Prime[n]]]]] (* Stefan Steinerberger, Feb 07 2006 *)
Select[Prime[Range[1500]],AllTrue[{#+2,Prime[PrimePi[#]+7]-2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 05 2019 *)

PARI

pnpk(n, m=7, k=2) = { local(x, v1, v2); for(x=1, n, v1 = prime(x)+k; v2 = prime(x+m)-k; if(isprime(v1)&isprime(v2), print1(v1-k, ", ") ) ) ; } \\ corrected by Amiram Eldar, Oct 04 2024

PARI

lista(pmax) = {my(k = 1, p = primes(8)); forprime(p1 = p[#p], pmax, k++; p[#p] = p1; if(p[2]- p[1] == 2 && p[8] - p[7] == 2, print1(p[1], ", ")); for(i = 1, #p-1, p[i] = p[i+1]));} \\ Amiram Eldar, Oct 04 2024