A333823 a(n) = Sum_{d|n, d odd} (n/d)^d.

1, 2, 4, 4, 6, 14, 8, 8, 37, 42, 12, 76, 14, 142, 384, 16, 18, 746, 20, 1044, 2552, 2070, 24, 536, 3151, 8218, 20440, 16412, 30, 41574, 32, 32, 178512, 131106, 94968, 263908, 38, 524326, 1596560, 32808, 42, 2379874, 44, 4194348, 16364502, 8388654, 48, 4144, 823593, 33654482

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, A333824.

Programs

Mathematica

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

PARI

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

Python

from sympy import divisors
def A333823(n): return sum((n//d)**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} k * x^k / (1 - k^2*x^(2*k)).

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