A339688 a(n) = Sum_{d|n} 8^(d-1).

1, 9, 65, 521, 4097, 32841, 262145, 2097673, 16777281, 134221833, 1073741825, 8589967945, 68719476737, 549756076041, 4398046515265, 35184374186505, 281474976710657, 2251799830495305, 18014398509481985, 144115188210078217, 1152921504607109185

Offset: 1

Author

Ilya Gutkovskiy, Dec 12 2020

Keywords

• nonn

Links

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

Crossrefs

Column 8 of A308813.

Cf. A001018, A320073.

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), this sequence (q=8), A339689 (q=9).

Programs

Magma

A339688:= func< n | (&+[8^(d-1): d in Divisors(n)]) >;
[A339688(n): n in [1..40]]; // G. C. Greubel, Jun 25 2024

Mathematica

Table[Sum[8^(d - 1), {d, Divisors[n]}], {n, 1, 21}]
nmax = 21; CoefficientList[Series[Sum[x^k/(1 - 8 x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

a(n) = sumdiv(n, d, 8^(d-1)); \\ Michel Marcus, Dec 13 2020

SageMath

def A339688(n): return sum(8^(k-1) for k in (1..n) if (k).divides(n))
[A339688(n) for n in range(1,41)] # G. C. Greubel, Jun 25 2024

Formula

G.f.: Sum_{k>=1} x^k / (1 - 8*x^k).

G.f.: Sum_{k>=1} 8^(k-1) * x^k / (1 - x^k).

a(n) ~ 8^(n-1). - Vaclav Kotesovec, Jun 05 2021