A333824 a(n) = Sum_{d|n, n/d odd} (n/d)^d.

1, 1, 4, 1, 6, 10, 8, 1, 37, 26, 12, 82, 14, 50, 384, 1, 18, 811, 20, 626, 2552, 122, 24, 6562, 3151, 170, 20440, 2402, 30, 74900, 32, 1, 178512, 290, 94968, 538003, 38, 362, 1596560, 390626, 42, 4901060, 44, 14642, 16364502, 530, 48, 43046722, 823593, 9766251

Offset: 1

Author

Ilya Gutkovskiy, Apr 06 2020

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A055225, A333823.

Programs

Mathematica

Table[DivisorSum[n, (n/#)^# &, OddQ[n/#] &], {n, 50}]
nmax = 50; CoefficientList[Series[Sum[(2 k - 1) x^(2 k - 1)/(1 - (2 k - 1) x^(2 k - 1)), {k, 1,nmax}], {x, 0, nmax}], x] // Rest

PARI

a(n) = sumdiv(n, d, if ((n/d)%2, (n/d)^d)); \\ Michel Marcus, Apr 07 2020

Python

from sympy import divisors
def A333824(n): return sum(d**(n//d) for d in divisors(n>>(~n & n-1).bit_length(),generator=True)) # Chai Wah Wu, Jul 09 2023

Formula

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

a(2^n) = 1. - Seiichi Manyama, Apr 07 2020

a(n) = Sum_{d|n, d odd} d^(n/d). - Chai Wah Wu, Jul 09 2023