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A259678 Primes of the form p(k)^2 + p(m)^2, where k and m are positive integers, and p(.) is the partition function given by A000041.

%I A259678 #13 Jan 25 2025 09:25:34
%S A259678 2,5,13,29,53,229,509,709,1021,1789,3137,3257,3361,6829,13337,18229,
%T A259678 30977,41177,49201,148229,240101,240109,250301,1004053,1575029,
%U A259678 2511601,3833989,3851989,5934121,6314389,9060109,9148309,13823549,20842361,31404937,106714213,116703973,151536109,221241901,221743789,230563789,254416549,323412629,323799793,332232629,336996053,468159773,676780229,676783361,682714321
%N A259678 Primes of the form p(k)^2 + p(m)^2, where k and m are positive integers, and p(.) is the partition function given by A000041.
%C A259678 The conjecture in A259531 implies that the current sequence has infinitely many terms.
%D A259678 Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
%H A259678 Zhi-Wei Sun, <a href="/A259678/b259678.txt">Table of n, a(n) for n = 1..10000</a>
%H A259678 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e A259678 a(1) = 2 since p(1)^2 + p(1)^2 = 2 is prime.
%e A259678 a(2) = 5 since p(1)^2 + p(2)^2 = 1^2 + 2^2 = 5 is prime.
%e A259678 a(3) = 13 since p(2)^2 + p(3)^2 = 2^2 + 3^2 = 13 is prime.
%e A259678 a(4) = 29 since p(2)^2 + p(4)^2 = 2^2 + 5^2 = 29 is prime.
%t A259678 p[n_]:=p[n]=PartitionsP[n];
%t A259678 tab={};Do[s=p[m]^2+p[k]^2;If[PrimeQ[s],tab=Append[tab,s]],{m,1,1600},{k,1,m}];
%t A259678 LL:=LL=Sort[DeleteDuplicates[tab]];Do[Print[n,LL[[n]]],{n,1,10000}]
%t A259678 Take[Select[Total/@Tuples[PartitionsP[Range[2000]]^2,2],PrimeQ]//Union,50] (* _Harvey P. Dale_, Jan 24 2025 *)
%Y A259678 Cf. A000040, A000041, A259531.
%K A259678 nonn
%O A259678 1,1
%A A259678 _Zhi-Wei Sun_, Jul 03 2015
%E A259678 Corrected and extended by _Harvey P. Dale_, Jan 24 2025