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A274609 Primes p such that both 2p-1 and 2p^2-2p+1 are prime.

%I A274609 #12 Dec 26 2022 16:14:51
%S A274609 2,3,31,331,1171,2011,2281,3181,4621,4861,6151,6211,6961,7951,8521,
%T A274609 9151,11251,12211,13411,15661,17491,18121,19141,20641,22531,23071,
%U A274609 23581,24631,25411,26041,26161,26431,26641,27091,27271,27361,27691,28201,28621,29221,31891,33151,34261,35491,36451
%N A274609 Primes p such that both 2p-1 and 2p^2-2p+1 are prime.
%C A274609 All values of a(n), except {2,3}, are equal to 1 mod 30.
%C A274609 These are also primes p such that both p^2+c and p^2-c are positive primes, for some c, when c is a square, since that requires c = (p-1)^2. Corresponding c values begin {1, 4, 900, 108900, ...}. This relates to a comment at A047222.
%H A274609 Charles R Greathouse IV, <a href="/A274609/b274609.txt">Table of n, a(n) for n = 1..10000</a>
%e A274609 31^2 - 30^2 = 61 and 31^2 + 30^2 = 1861 are both prime.
%t A274609 result = {}; Do[If[PrimeQ[2*Prime[i] - 1] && PrimeQ[2*Prime[i]^2 - 2*Prime[i] + 1], AppendTo[result, Prime[i]]], {i, 1, 10000}]; result
%t A274609 Select[Prime[Range[4000]],AllTrue[{2#-1,2#^2-2#+1},PrimeQ]&] (* _Harvey P. Dale_, Dec 26 2022 *)
%o A274609 (PARI) is(n)=isprime(2*n-1) && isprime(2*n^2-2*n+1) && isprime(n) \\ _Charles R Greathouse IV_, Jul 15 2016
%Y A274609 Cf. A047222.
%K A274609 nonn
%O A274609 1,1
%A A274609 _Richard R. Forberg_, Jun 30 2016