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

1,2

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.

Links

Table of n, a(n) for n=1..88.

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.

Example

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.

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

Crossrefs

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

Sequence in context: A257727 A039273 A039164 * A335040 A270427 A270189

Adjacent sequences: A263974 A263975 A263976 * A263978 A263979 A263980

Keyword

nonn

Author

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

Status

approved