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A292919 Sum of n-th powers of odd divisors of n.

%I A292919 #17 Sep 08 2018 02:47:53
%S A292919 1,1,28,1,3126,730,823544,1,387440173,9765626,285311670612,531442,
%T A292919 302875106592254,678223072850,437893920912786408,1,
%U A292919 827240261886336764178,150094635684419611,1978419655660313589123980,95367431640626,5842587018944528395924761632,81402749386839761113322
%N A292919 Sum of n-th powers of odd divisors of n.
%H A292919 Seiichi Manyama, <a href="/A292919/b292919.txt">Table of n, a(n) for n = 1..386</a>
%H A292919 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F A292919 a(n) = [x^n] Sum_{k>=1} (2*k - 1)^n*x^(2*k-1)/(1 - x^(2*k-1)).
%F A292919 a(2^k) = 1.
%p A292919 f:= proc(n) local t,d;
%p A292919   t:= n/2^padic:-ordp(n,2);
%p A292919   add(d^n, d = numtheory:-divisors(t));
%p A292919 end proc:
%p A292919 map(f, [$1..30]); # _Robert Israel_, Sep 27 2017
%t A292919 Rest[Table[SeriesCoefficient[Sum[(2 k - 1)^n x^(2 k - 1)/(1 - x^(2 k - 1)), {k, 1, n}], {x, 0, n}], {n, 0, 22}]]
%t A292919 f[n_] := Plus @@ (Select[Divisors[n], OddQ]^n); Array[f, 22] (* _Robert G. Wilson v_, Sep 26 2017 *)
%o A292919 (PARI) a(n) = sumdiv(n, d, if (d%2, d^n)); \\ _Michel Marcus_, Sep 08 2018
%Y A292919 Diagonal of A285425.
%Y A292919 Cf. A000593, A023887, A082245.
%K A292919 nonn,look
%O A292919 1,3
%A A292919 _Ilya Gutkovskiy_, Sep 26 2017