A357041 a(n) = Sum_{d|n} 2^(d-1) * binomial(d+n/d-1,d).

1, 4, 7, 18, 21, 66, 71, 196, 305, 648, 1035, 2526, 4109, 8774, 16875, 34288, 65553, 134860, 262163, 531506, 1051237, 2109594, 4194327, 8425348, 16779257, 33611984, 67123631, 134350206, 268435485, 537178750, 1073741855, 2148064768, 4295048345, 8591114580

Offset: 1

Author

Seiichi Manyama, Feb 26 2023

Keywords

• nonn

Crossrefs

Cf. A081543, A338682.

Cf. A360797.

Programs

Mathematica

a[n_] := DivisorSum[n, 2^(#-1) * Binomial[# + n/# - 1, #] &]; Array[a, 50] (* Amiram Eldar, Jul 31 2023 *)

PARI

a(n) = sumdiv(n, d, 2^(d-1)*binomial(d+n/d-1, d));

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (1/(1-2*x^k)^k-1))/2)

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (2*x)^k/(1-x^k)^(k+1))/2)

Python

from math import comb
from sympy import divisors
def A357041(n): return sum(comb(d+n//d-1,d)<Chai Wah Wu, Feb 27 2023

Formula

G.f.: (1/2) * Sum_{k>0} (1/(1 - 2 * x^k)^k - 1).

G.f.: (1/2) * Sum_{k>0} (2 * x)^k/(1 - x^k)^(k+1).

If p is prime, a(p) = p + 2^(p-1).