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%I #12 Dec 14 2019 08:07:43
%S 1,5,45,218,506,538,1027
%N Numbers n such that (Product of first n twin prime pair averages [A014574] ) + 1 is prime.
%C a(3) and a(4) certified prime with Primo.
%e a(2) = 5 because the product of the first five terms of A014574 + 1, 4*6*12*18*30 + 1 = 155521, is prime.
%t seq = {1}; prod = 4; k = 1; Do[If[AllTrue[6 n + {-1, 1}, PrimeQ], prod *= 6 n; k++; If[PrimeQ[prod + 1], AppendTo[seq, k]]], {n, 1, 5000}]; seq (* _Amiram Eldar_, Dec 14 2019 *)
%o (PARI) { atp(N)= local(t, c); t=1; c=0; forprime(p=2,N,if(isprime(p+2), t*=p+1; c++; if(isprime(t+1), print1(c, ", ")))) }
%Y Cf. A014574.
%K nonn,more
%O 1,2
%A _Jason Earls_, Dec 26 2002
%E a(7) from _Amiram Eldar_, Dec 14 2019
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