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%I #18 Nov 07 2015 17:03:06
%S 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,
%T 34,35,36,37,38,40,42,44,45,46,47,48,50,52,54,56,57,58,60,62,64,65,66,
%U 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
%N Integers k > 0 such that k^2 + p^2 is prime for some prime p.
%C The smallest such prime p is in A263726.
%C Complement of A263722.
%C An odd number k is a member if and only if k^2 + 4 is prime; see A007591.
%C 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.
%H Stephan Baier and Liangyi Zhao, <a href="http://arxiv.org/abs/math/0703284">On Primes Represented by Quadratic Polynomials</a>, Anatomy of Integers, CRM Proc. & Lecture Notes, Vol. 46, Amer. Math. Soc. 2008, pp. 169 - 166.
%H Étienne Fouvry and Henryk Iwaniec, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa79/aa7935.pdf">Gaussian primes</a>, Acta Arithmetica 79:3 (1997), pp. 249-287.
%e 1^2 + 2^2 = 5, and 2 and 5 are prime, so a(1) = 1.
%e 9^2 + p^2 is composite for all primes p, so 9 is not a member.
%t 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
%Y Cf. A007591, A240130, A240131, A263466, A263722, A263726.
%K nonn
%O 1,2
%A _Jonathan Sondow_ and _Robert G. Wilson v_, Oct 30 2015
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