A248036 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.

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, 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, 32, 33, 5, 24, 7, 59, 58, 15

Offset: 1

Author

Zhi-Wei Sun, Sep 29 2014

Keywords

• nonn

Comments

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

Examples

			a(5) = 10 since 10 + 5 = 15 divides sigma(10)^2 + sigma(5)^2 = 18^2 + 6^2 = 360.
a(1024) = 2098177 since 2098177 + 1024 = 2099201 divides sigma(2098177)^2 + sigma(1024)^2 = 2103300^2 + 2047^2 = 4423875080209 = 2099201*2107409.
		

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, A247975, A248008, A248035.

Programs

Mathematica

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}]
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 *)