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A327581 a(1) is the smallest prime p such that 6*p^2-1 and 6*p^2+1 are twin primes; for n > 1, a(n) is the smallest prime q > a(n-1) such that 6*q^prime(n)-1 and 6*q^prime(n)+1 are twin primes or 0 if no solution exists.

%I A327581 #24 Oct 21 2019 14:10:00
%S A327581 5,0,2557,51137,52057,55373,88867,95273,179947,236653,993647,1010467,
%T A327581 1935533,2031767,2138803,2849317,8031343,11696563,11715133,18125993,
%U A327581 22615493,26766633,26801393,29963077,39377893,58282927,70354657,98988257,119772847,141442493,145460123
%N A327581 a(1) is the smallest prime p such that 6*p^2-1 and 6*p^2+1 are twin primes; for n > 1, a(n) is the smallest prime q > a(n-1) such that 6*q^prime(n)-1 and 6*q^prime(n)+1 are twin primes or 0 if no solution exists.
%C A327581 For prime(2) = 3 there is no solution such that 6*q^3-1 and 6*q^3+1 with q prime are twin primes. Because 7 divides 6*p^3-1 when p == 3, 5, 6 mod 7, 7 divides 6*p^3+1 when p == 1, 2, 4 mod 7. Therefore p can only be 7. But then 6*7^3-1 = 11^2*17 and 6*7^3+1 = 29*71 are not prime numbers, so a(2)=0.
%H A327581 Pierre CAMI, <a href="/A327581/a327581.txt">PFGW Script</a>
%o A327581 (PARI) findp(n, pmin) = {my(pmin = nextprime(pmin+1), q); forprime(p=pmin, , if (isprime(q=6*p^prime(n)-1) && isprime(q+2), return(p));); }
%o A327581 lista(nn) = {my(lasta = 2, newa); print1(findp(1, lasta), ", 0"); for (n=3, nn, newa = findp(n, lasta); print1(", ", newa); lasta = newa;); } \\ _Michel Marcus_, Sep 20 2019
%K A327581 nonn
%O A327581 1,1
%A A327581 _Pierre CAMI_, Sep 17 2019