A339689 - OEIS

%I #18 Jun 26 2024 02:05:59

%S 1,10,82,739,6562,59140,531442,4783708,43046803,387427060,3486784402,

%T 31381119478,282429536482,2541866359780,22876792461604,

%U 205891136878357,1853020188851842,16677181742772430,150094635296999122,1350851718060419878,12157665459057460324

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

%H Seiichi Manyama, <a href="/A339689/b339689.txt">Table of n, a(n) for n = 1..1000</a>

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

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

%F a(n) ~ 9^(n-1). - _Vaclav Kotesovec_, Jun 05 2021

%t Table[Sum[9^(d - 1), {d, Divisors[n]}], {n, 1, 21}]

%t nmax = 21; CoefficientList[Series[Sum[x^k/(1 - 9 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%o (PARI) a(n) = sumdiv(n, d, 9^(d-1)); \\ _Michel Marcus_, Dec 13 2020

%o (Magma)

%o A339689:= func< n | (&+[9^(d-1): d in Divisors(n)]) >;

%o [A339689(n): n in [1..40]]; // _G. C. Greubel_, Jun 25 2024

%o (SageMath)

%o def A339689(n): return sum(9^(k-1) for k in (1..n) if (k).divides(n))

%o [A339689(n) for n in range(1,41)] # _G. C. Greubel_, Jun 25 2024

%Y Column 9 of A308813.

%Y Cf. A001019, A320074.

%Y 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).

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Dec 12 2020