A333694 Expansion of Sum_{k>=1} k * x^k / (1 - x^(k^2)).

1, 3, 4, 5, 6, 9, 8, 9, 10, 13, 12, 16, 14, 17, 16, 17, 18, 21, 20, 25, 25, 25, 24, 25, 26, 29, 28, 29, 30, 41, 32, 33, 34, 37, 36, 41, 38, 41, 43, 41, 42, 51, 44, 45, 46, 49, 48, 52, 50, 53, 52, 57, 54, 57, 61, 64, 61, 61, 60, 61, 62, 65, 64, 65, 66, 72, 68, 73, 70, 73

Offset: 1

Author

Ilya Gutkovskiy, Apr 04 2020

Keywords

• nonn

Comments

Sum of divisors d of n such that n/d == 1 (mod d).

Crossrefs

Cf. A000203, A069290, A163671.

Programs

Maple

a:= n-> add(`if`(irem(n/d-1, d)=0, d, 0), d=numtheory[divisors](n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Apr 04 2020

Mathematica

nmax = 70; CoefficientList[Series[Sum[k x^k/(1 - x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, Mod[n/# - 1, #] == 0 &], {n, 1, 70}]

PARI

A333694(n) = sumdiv(n,d,d*(0==(((n/d)-1)%d))); \\ Antti Karttunen, Apr 04 2020, after the second Mathematica program.

Formula

G.f.: Sum_{k>=1} k * x^k / (1 - x^(k^2)).