A248008 Least positive integer m such that m + n divides sigma(m*n), where sigma(k) denotes the sum of all positive divisors of k.

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

Offset: 1

Author

Zhi-Wei Sun, Sep 29 2014

Keywords

• nonn

Comments

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

a(n) = 1 if and only if n is in A230606. Also, if a(i) = j, a(j) <= i. - Derek Orr, Sep 29 2014

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

Examples

			a(6) = 4 since 4 + 6 = 10 divides sigma(4*6) = 60.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Crossrefs

Cf. A000203, A248004, A248006, A248007.

Programs

Mathematica

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}]

PARI

a(n)=m=1;while(sigma(m*n)%(m+n),m++);m
vector(100,n,a(n)) \\ Derek Orr, Sep 29 2014