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%I #61 Jul 18 2015 14:52:31
%S 1,1,14,11,6,31,2,34,2,76,1,100,71,38,1,7,62,1128,1,180,123,15,174,
%T 128,4,111,110,2,4,2,2241,21,144,416,397,31,11,8,15,5,91,56,53,23,89,
%U 18,25,341,12,1,66,454,159,36,573,26,2,488,72,416,802,440,28,30,595,17,236,947,1289,1287,1000,367,80,407,1,77,938,150,36,1
%N Least positive integer k such that p(k)^2 + p(k*n)^2 is prime, where p(.) is the partition function given by A000041, or 0 if no such k exists.
%C Conjecture: Any positive rational number r can be written as m/n, where m and n are positive integers with p(m)^2 + p(n)^2 prime.
%C For example, 4/5 = 124/155, and the number p(124)^2 + p(155)^2 = 2841940500^2 + 66493182097^2 = 4429419891190341567409 is prime.
%C We also guess that any positive rational number can be written as m/n, where m and n are positive integers with p(m)+p(n) (or p(m)*p(n)-1, or p(m)*p(n)+1) prime.
%D 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 Zhi-Wei Sun, <a href="/A259531/b259531.txt">Table of n, a(n) for n = 1..200</a>
%H Zhi-Wei Sun, <a href="/A259531/a259531_4.txt">Checking the conjecture for r = a/b with a,b = 1..100</a>
%H 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 a(5) = 6 since p(6)^2 + p(6*5)^2 = 11^2 + 5604^2 = 31404937 is prime.
%t Do[k=0;Label[bb];k=k+1;If[PrimeQ[PartitionsP[k]^2+PartitionsP[k*n]^2],Goto[aa],Goto[bb]];Label[aa];Print[n," ",k]; Continue,{n,1,80}]
%Y Cf. A000040, A000041, A258803, A258836, A259487, A259492, A259540.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Jul 02 2015
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