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

1,1

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.

Links

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A047222.

Sequence in context: A347332 A054551 A049065 * A372142 A128030 A239593

Adjacent sequences: A274606 A274607 A274608 * A274610 A274611 A274612

Keyword

nonn

Author

Richard R. Forberg, Jun 30 2016

Status

approved