A339684 a(n) = Sum_{d|n} 4^(d-1).

1, 5, 17, 69, 257, 1045, 4097, 16453, 65553, 262405, 1048577, 4195413, 16777217, 67112965, 268435729, 1073758277, 4294967297, 17179935765, 68719476737, 274878169413, 1099511631889, 4398047559685, 17592186044417, 70368748389461, 281474976710913

Offset: 1

Author

Ilya Gutkovskiy, Dec 12 2020

Keywords

• nonn

Links

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

Crossrefs

Cf. A000302, A295505.

Sums of the form Sum_{d|n} q^(d-1): A034729 (q=2), A034730 (q=3), A113999 (q=10), this sequence (q=4), A339685 (q=5), A339686 (q=6), A339687 (q=7), A339688 (q=8), A339689 (q=9).

Programs

Magma

A339684:= func< n | (&+[4^(d-1): d in Divisors(n)]) >;
[A339684(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Mathematica

Table[Sum[4^(d - 1), {d, Divisors[n]}], {n, 1, 25}]
nmax = 25; CoefficientList[Series[Sum[x^k/(1 - 4 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

a(n) = sumdiv(n, d, 4^(d-1)); \\ Michel Marcus, Dec 13 2020

SageMath

def A339684(n): return sum(4^(k-1) for k in (1..n) if (k).divides(n))
[A339684(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

Formula

G.f.: Sum_{k>=1} x^k / (1 - 4*x^k).

G.f.: Sum_{k>=1} 4^(k-1) * x^k / (1 - x^k).

a(n) ~ 4^(n-1). - Vaclav Kotesovec, Jun 05 2021