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%I #13 Oct 03 2014 04:57:34
%S 1,1,3,1,6,1,7,4,8,1,2,1,10,9,15,1,8,1,1,11,14,1,6,6,16,5,14,1,6,1,10,
%T 15,11,13,16,1,7,9,5,1,6,1,12,7,26,1,14,8,12,21,46,1,6,17,4,23,32,1,
%U 24,1,34,41,63,7,6,1,16,11,2
%N Least positive integer m such that m + n divides phi(m)*sigma(n), where phi(.) and sigma(.) are given by A000010 and A000203.
%C Conjecture: For any n > 1, we have a(n) <= n.
%C The existence of a(n) is easy; in fact, for m = sigma(n) - n, obviously m + n divides phi(m)*sigma(n). - _Zhi-Wei Sun_, Oct 02 2014
%H Zhi-Wei Sun, <a href="/A248029/b248029.txt">Table of n, a(n) for n = 2..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(8) = 7 since 7 + 8 = 15 divides phi(7)*sigma(8) = 6*15 = 90.
%t Do[m=1;Label[aa];If[Mod[EulerPhi[m]*DivisorSigma[1,n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,2,70}]
%o (PARI)
%o a(n)=m=1;while((eulerphi(m)*sigma(n))%(m+n),m++);m
%o vector(100,n,a(n)) \\ _Derek Orr_, Sep 29 2014
%Y Cf. A000010, A000203, A248004, A248007, A248008, A248030.
%K nonn
%O 2,3
%A _Zhi-Wei Sun_, Sep 29 2014
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