A339689 a(n) = Sum_{d|n} 9^(d-1).

1, 10, 82, 739, 6562, 59140, 531442, 4783708, 43046803, 387427060, 3486784402, 31381119478, 282429536482, 2541866359780, 22876792461604, 205891136878357, 1853020188851842, 16677181742772430, 150094635296999122, 1350851718060419878, 12157665459057460324

Offset: 1

Author

Ilya Gutkovskiy, Dec 12 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000

Crossrefs

Column 9 of A308813.

Cf. A001019, A320074.

Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), this sequence (q=9).

Programs

Magma

A339689:= func< n | (&+[9^(d-1): d in Divisors(n)]) >;
[A339689(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Mathematica

Table[Sum[9^(d - 1), {d, Divisors[n]}], {n, 1, 21}]
nmax = 21; CoefficientList[Series[Sum[x^k/(1 - 9 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

a(n) = sumdiv(n, d, 9^(d-1)); \\ Michel Marcus, Dec 13 2020

SageMath

def A339689(n): return sum(9^(k-1) for k in (1..n) if (k).divides(n))
[A339689(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

Formula

G.f.: Sum_{k>=1} x^k / (1 - 9*x^k).

G.f.: Sum_{k>=1} 9^(k-1) * x^k / (1 - x^k).

a(n) ~ 9^(n-1). - Vaclav Kotesovec, Jun 05 2021