A034730 Dirichlet convolution of b_n=1 with c_n=3^(n-1).

1, 4, 10, 31, 82, 256, 730, 2218, 6571, 19768, 59050, 177430, 531442, 1595056, 4783060, 14351125, 43046722, 129146980, 387420490, 1162281262, 3486785140, 10460412256, 31381059610, 94143358444, 282429536563, 847289140888, 2541865834900, 7625599080070, 22876792454962

Offset: 1

Author

Erich Friedman

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..2000

Crossrefs

Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), this sequence (q=3), A113999 (q=10), A339684 (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

Programs

Magma

A034730:= func< n | (&+[3^(d-1): d in Divisors(n)]) >;
[A034730(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Mathematica

Rest[CoefficientList[Series[Sum[x^k/(1-3*x^k),{k,1,30}],{x,0,30}],x]] (* Vaclav Kotesovec, Sep 09 2014 *)
A034730[n_]:= DivisorSum[n, 3^(#-1) &];
Table[A034730[n], {n,40}] (* G. C. Greubel, Jun 25 2024 *)

PARI

{a(n) = sumdiv(n, d, 3^(d-1))} \\ Seiichi Manyama, Jun 26 2019

SageMath

def A034730(n): return sum(3^(k-1) for k in (1..n) if (k).divides(n))
[A034730(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

Formula

G.f.: Sum_{n>0} x^n/(1-3*x^n). - Vladeta Jovovic, Nov 14 2002

a(n) ~ 3^(n-2). - Vaclav Kotesovec, Sep 09 2014

a(n) = Sum_{d|n} 3^(d-1). - Seiichi Manyama, Jun 26 2019