A263977 Integers k > 0 such that k^2 + p^2 is prime for some prime p.

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 15, 16, 17, 18, 20, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 37, 38, 40, 42, 44, 45, 46, 47, 48, 50, 52, 54, 56, 57, 58, 60, 62, 64, 65, 66, 67, 68, 70, 72, 73, 74, 76, 78, 80, 82, 84, 85, 86, 87, 88, 90, 92, 94, 95, 96, 97, 98, 100, 102, 103, 104, 106, 108, 110, 112, 114, 115, 116, 117, 118, 120, 122, 124, 125, 126

Offset: 1

Author

Jonathan Sondow and Robert G. Wilson v, Oct 30 2015

Keywords

• nonn

Comments

The smallest such prime p is in A263726.

Complement of A263722.

An odd number k is a member if and only if k^2 + 4 is prime; see A007591.

Conjecture: Every even number k is a member. (This is equivalent to the Conjecture in A263722.) We have checked this for all k <= 12*10^7.

Examples

			1^2 + 2^2 = 5, and 2 and 5 are prime, so a(1) = 1.
9^2 + p^2 is composite for all primes p, so 9 is not a member.
		

Links

• Stephan Baier and Liangyi Zhao, On Primes Represented by Quadratic Polynomials, Anatomy of Integers, CRM Proc. & Lecture Notes, Vol. 46, Amer. Math. Soc. 2008, pp. 169 - 166.
• Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica 79:3 (1997), pp. 249-287.

Crossrefs

Cf. A007591, A240130, A240131, A263466, A263722, A263726.

Programs

Mathematica

fQ[n_] := Block[{p = 2}, While[ !PrimeQ[n^2 + p^2] && p < 1500, p = NextPrime@ p]; If[p > 1500, 0, p]]; lst = {}; k = 1; While[k < 130, If[fQ@ k > 0, AppendTo[lst, k]]; k++]; lst