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%I #19 Aug 05 2019 09:26:25
%S 2,5,13,17,29,37,41,53,61,73,89,101,109,113,137,149,157,193,229,241,
%T 349,373,509,709,733,1033,1049,1213,1249,1453,1493,1669,1789,2141,
%U 2237,2341,2917,3037,3137,3361,4217,5801,5897,6029,6073,8821,10301,10937,11057,18229,18289,19249,20173,20341,20389,21017,24001,30977,36913,42793
%N Primes of the form p(k)^2 + q(m)^2 with k > 0 and m > 0, where p(.) is the partition function (A000041), and q(.) is the strict partition function (A000009).
%C Conjecture: The sequence contains infinitely many terms.
%C This follows from part (i) of the conjecture in A223307. Similarly, the conjecture in A232504 implies that there are infinitely many primes of the form p(k) + q(m) with k and m positive integers.
%H Zhi-Wei Sun, <a href="/A233346/b233346.txt">Table of n, a(n) for n = 1..176</a>
%H Z.-W. Sun, <a href="http://arxiv.org/abs/1312.1166">On a^n+ bn modulo m</a>, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
%e a(1) = 2 since p(1)^2 + q(1)^2 = 1^2 + 1^2 = 2.
%e a(2) = 5 since p(1)^2 + q(3)^2 = 1^2 + 2^2 = 5.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t n=0
%t Do[If[Mod[Prime[m]+1,4]>0,Do[If[PartitionsP[j]>=Sqrt[Prime[m]],Goto[aa],
%t If[SQ[Prime[m]-PartitionsP[j]^2]==False,Goto[bb],Do[If[PartitionsQ[k]^2==Prime[m]-PartitionsP[j]^2,
%t n=n+1;Print[n," ",Prime[m]];Goto[aa]];If[PartitionsQ[k]^2>Prime[m]-PartitionsP[j]^2,Goto[bb]];Continue,{k,1,2*Sqrt[Prime[m]]}]]];
%t Label[bb];Continue,{j,1,Sqrt[Prime[m]]}]];
%t Label[aa];Continue,{m,1,4475}]
%Y Cf. A000009, A000040, A000041, A002313, A232504, A233307.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Dec 07 2013
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