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%I #14 Oct 02 2014 05:40:22
%S 1,2,1,8,7,7,5,1,25,35,15,14,13,12,1,4,23,532,22,385,113,1,17,138,8,
%T 92,80,44,116,128,586,165,5,464,10,39,80,38,1,52,33,118,6,28,11,1239,
%U 47,92,517,3,145,40,8,184,104,104,16,73,53,52,5,145,172,68,11
%N Least positive integer m such that m + n divides pi(m^2) + pi(n^2), where pi(x) denotes the number of primes not exceeding x.
%C Conjecture: (i) a(n) exists for any n > 0.
%C (ii) For each integer m > 0, there is a positive integer n such that m + n divides prime(m^2) + prime(n^2).
%H Zhi-Wei Sun, <a href="/A248052/b248052.txt">Table of n, a(n) for n = 1..1000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1409.5685">A new theorem on the prime-counting function</a>, arXiv:1409.5685, 2014.
%e a(4) = 8 since 8 + 4 = 12 divides pi(8^2) + pi(4^2) = 18 + 6 = 24.
%t Do[m=1;Label[aa];If[Mod[PrimePi[m^2]+PrimePi[n^2],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,70}]
%t a[ n_] := If[ n<1, 0, Module[ {m=1}, While[ Not @ Divisible[ PrimePi[m^2] + PrimePi[n^2], m + n], m++]; m]]; (* _Michael Somos_, Sep 30 2014 *)
%Y Cf. A000040, A000720, A247824, A247975, A248044, A248054.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Sep 30 2014
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