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

2, 3, 31, 331, 1171, 2011, 2281, 3181, 4621, 4861, 6151, 6211, 6961, 7951, 8521, 9151, 11251, 12211, 13411, 15661, 17491, 18121, 19141, 20641, 22531, 23071, 23581, 24631, 25411, 26041, 26161, 26431, 26641, 27091, 27271, 27361, 27691, 28201, 28621, 29221, 31891, 33151, 34261, 35491, 36451

Offset: 1

Author

Richard R. Forberg, Jun 30 2016

Keywords

• nonn

Comments

All values of a(n), except {2,3}, are equal to 1 mod 30.

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.

Examples

			31^2 - 30^2 = 61 and 31^2 + 30^2 = 1861 are both prime.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A047222.

Programs

Mathematica

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
Select[Prime[Range[4000]],AllTrue[{2#-1,2#^2-2#+1},PrimeQ]&] (* Harvey P. Dale, Dec 26 2022 *)

PARI

is(n)=isprime(2*n-1) && isprime(2*n^2-2*n+1) && isprime(n) \\ Charles R Greathouse IV, Jul 15 2016