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A263977 Integers k > 0 such that k^2 + p^2 is prime for some prime p. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 6 08:31 EST 2022. Contains 358605 sequences. (Running on oeis4.)