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%I #10 May 07 2016 17:49:25
%S 1,3,2,1,10,6,3,50,1,5,34,28,7,6,10,18,3,16,33,5,20,14,83,24,1,10,10,
%T 12,56,6,33,2,15,11,93,13,204,27,52,38,17,6,7,6,15,14,5,944,1,8,17,39,
%U 32,33,5,24,7,59,58,15
%N Least positive integer m such that m + n divides sigma(m)^2 + sigma(n)^2, where sigma(k) denotes the number of positive divisors of k.
%C Conjecture: a(n) exists for any n > 0.
%H Zhi-Wei Sun, <a href="/A248036/b248036.txt">Table of n, a(n) for n = 1..10000</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(5) = 10 since 10 + 5 = 15 divides sigma(10)^2 + sigma(5)^2 = 18^2 + 6^2 = 360.
%e a(1024) = 2098177 since 2098177 + 1024 = 2099201 divides sigma(2098177)^2 + sigma(1024)^2 = 2103300^2 + 2047^2 = 4423875080209 = 2099201*2107409.
%t Do[m=1;Label[aa];If[Mod[DivisorSigma[1,m]^2+DivisorSigma[1,n]^2,m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]
%t lpi[n_]:=Module[{m=1,dsn=DivisorSigma[1,n]^2},While[ !Divisible[ DivisorSigma[ 1,m]^2+ dsn, m+n], m++];m]; Array[lpi, 60] (* _Harvey P. Dale_, May 07 2016 *)
%Y Cf. A000203, A247975, A248008, A248035.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Sep 29 2014
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