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

1,2

Comments

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

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.

Example

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.

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 *)

Crossrefs

Cf. A000203, A247975, A248008, A248035.

Sequence in context: A267278 A267019 A100100 * A307214 A185967 A188111

Adjacent sequences: A248033 A248034 A248035 * A248037 A248038 A248039

Keyword

nonn

Author

Zhi-Wei Sun, Sep 29 2014

Status

approved