%I #12 Sep 05 2019 09:30:34
%S 17,71,149,191,431,521,821,1049,1277,1289,1451,1619,1667,1877,1949,
%T 2027,2657,3299,3329,3467,3527,3539,3767,3929,4271,4931,5477,5849,
%U 6131,6659,6701,6779,6827,8537,8819,8999,9419,9719,9929,10037,10091,11069,11117
%N Numbers p(n) such that p(n)+2 and p(n+7)2 are both prime numbers, where p(n) is the nth prime.
%C Conjecture: There are an infinite number of primes p(n) such that p(n)2 and p(n+k)2 are both prime for all k > 1.
%H Harvey P. Dale, <a href="/A105414/b105414.txt">Table of n, a(n) for n = 1..1000</a>
%e p(8)2 = 17, p(8+6)2 = 41, both prime, 17 is in the sequence.
%t 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 *)
%t Select[Prime[Range[1500]],AllTrue[{#+2,Prime[PrimePi[#]+7]2},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Sep 05 2019 *)
%o (PARI) pnpk(n,m,k) = \ both are prime { local(x,l1,l2,v1,v2); for(x=1,n, v1 = prime(x)+ k; v2 = prime(x+m)+k; if(isprime(v1)&isprime(v2), \ print1(x",") print1(v1",") ) ) }
%K nonn
%O 1,1
%A _Cino Hilliard_, May 02 2005
