A333824 - OEIS

%I #16 Jul 09 2023 12:13:29

%S 1,1,4,1,6,10,8,1,37,26,12,82,14,50,384,1,18,811,20,626,2552,122,24,

%T 6562,3151,170,20440,2402,30,74900,32,1,178512,290,94968,538003,38,

%U 362,1596560,390626,42,4901060,44,14642,16364502,530,48,43046722,823593,9766251

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

%H Seiichi Manyama, <a href="/A333824/b333824.txt">Table of n, a(n) for n = 1..5000</a>

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

%F a(2^n) = 1. - _Seiichi Manyama_, Apr 07 2020

%F a(n) = Sum_{d|n, d odd} d^(n/d). - _Chai Wah Wu_, Jul 09 2023

%t Table[DivisorSum[n, (n/#)^# &, OddQ[n/#] &], {n, 50}]

%t 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

%o (PARI) a(n) = sumdiv(n, d, if ((n/d)%2, (n/d)^d)); \\ _Michel Marcus_, Apr 07 2020

%o (Python)

%o from sympy import divisors

%o 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

%Y Cf. A055225, A333823.

%K nonn

%O 1,3

%A _Ilya Gutkovskiy_, Apr 06 2020