A363913 a(n) = Sum_{k=0..n} divides(k, n) * 3^k, where divides(k, n) = 1 if k divides n, otherwise 0.

1, 3, 12, 30, 93, 246, 768, 2190, 6654, 19713, 59304, 177150, 532290, 1594326, 4785168, 14349180, 43053375, 129140166, 387440940, 1162261470, 3486843786, 10460355420, 31381236768, 94143178830, 282430075332, 847288609689, 2541867422664, 7625597504700, 22876797240210

Offset: 0

Author

Peter Luschny, Jun 28 2023

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000007 (m = 0), A000005 (m = 1), A055895 (m = 2), this sequence (m = 3).

Cf. A113704, A363733, A363734, A363735, A363421.

Programs

Magma

A363913:= func< n | n eq 0 select 1 else 3*(&+[3^(d-1): d in Divisors(n)]) >;
[A363913(n): n in [0..40]]; // G. C. Greubel, Jun 26 2024

Maple

divides := (k, n) -> ifelse(k = n or (k > 0 and irem(n, k) = 0), 1, 0):
a := n -> local j; add(divides(j, n) * 3^j, j = 0 ..n): seq(a(n), n = 0..28);

Mathematica

A363913[n_]:= If[n==0, 1, 3*DivisorSum[n, 3^(#-1) &]];
Table[A363913[n], {n,0,40}] (* G. C. Greubel, Jun 26 2024 *)

Python

from sympy import divisors
def A363913(n): return sum(3**k for k in divisors(n,generator=True)) if n else 1 # Chai Wah Wu, Jun 28 2023

SageMath

def a(n): return sum(3^k * k.divides(n) for k in srange(n+1))
print([a(n) for n in range(29)])

Formula

a(n) = Sum_{j=0..n} A113704(j, n) * m^j for m = 3; for other cases see the crossreferences.

a(n) = 3*A034730(n), n>=1. - R. J. Mathar, Jul 04 2023