A327629 Expansion of Sum_{k>=1} x^(k*(k + 1)/2) / (1 - x^(k*(k + 1)/2))^2.

1, 2, 4, 4, 5, 9, 7, 8, 12, 11, 11, 18, 13, 14, 21, 16, 17, 27, 19, 22, 29, 22, 23, 36, 25, 26, 36, 29, 29, 50, 31, 32, 44, 34, 35, 55, 37, 38, 52, 44, 41, 65, 43, 44, 64, 46, 47, 72, 49, 55, 68, 52, 53, 81, 56, 58, 76, 58, 59, 100, 61, 62, 87, 64, 65, 100, 67, 68, 92, 77

Offset: 1

Author

Ilya Gutkovskiy, Sep 19 2019

Keywords

• nonn

Comments

Sum of divisors d of n such that n/d is triangular number.

Crossrefs

Cf. A000217, A006463, A007862, A010054, A076752, A112886 (fixed points), A185027.

Programs

Mathematica

nmax = 70; CoefficientList[Series[Sum[x^(k (k + 1)/2)/(1 - x^(k (k + 1)/2))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, # &, IntegerQ[Sqrt[8 n/# + 1]] &]; Table[a[n], {n, 1, 70}]

PARI

a(n)={sumdiv(n, d, if(ispolygonal(d,3), n/d))} \\ Andrew Howroyd, Sep 19 2019

Python

from sympy import divisors
from sympy.ntheory.primetest import is_square
def A327629(n): return sum(n//d for d in divisors(n,generator=True) if is_square((d<<3)+1)) # Chai Wah Wu, Jun 07 2025

Formula

a(n) = Sum_{d|n} A010054(n/d) * d.