A248008 - OEIS

%I #11 Oct 03 2014 04:58:41

%S 2,1,1,3,1,4,1,7,4,14,1,18,1,10,9,15,1,12,1,1,11,5,1,4,6,4,6,2,1,18,1,

%T 28,6,14,13,13,1,12,17,22,1,22,1,10,3,10,1,30,8,12,9,18,1,2,17,6,7,26,

%U 1,52,1,22,28,38,19,12,1,22,36,26

%N Least positive integer m such that m + n divides sigma(m*n), where sigma(k) denotes the sum of all positive divisors of k.

%C Conjecture: a(n) exists for any n > 0.

%C a(n) = 1 if and only if n is in A230606. Also, if a(i) = j, a(j) <= i. - _Derek Orr_, Sep 29 2014

%C Numbers n such that a(n) > n: 1, 10, 12, 108, 1139, ... The next number, if it exists, is greater than 2*10^4. - _Derek Orr_, Sep 29 2014

%H Zhi-Wei Sun, <a href="/A248008/b248008.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(6) = 4 since 4 + 6 = 10 divides sigma(4*6) = 60.

%t Do[m=1; Label[aa]; If[Mod[DivisorSigma[1,m*n], m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n,1,70}]

%o (PARI)

%o a(n)=m=1;while(sigma(m*n)%(m+n),m++);m

%o vector(100,n,a(n)) \\ _Derek Orr_, Sep 29 2014

%Y Cf. A000203, A248004, A248006, A248007.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Sep 29 2014