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A352032 Sum of the 4th powers of the odd proper divisors of n.

%I A352032 #20 Oct 11 2023 03:54:09
%S A352032 0,1,1,1,1,82,1,1,82,626,1,82,1,2402,707,1,1,6643,1,626,2483,14642,1,
%T A352032 82,626,28562,6643,2402,1,51332,1,1,14723,83522,3027,6643,1,130322,
%U A352032 28643,626,1,196964,1,14642,57893,279842,1,82,2402,391251,83603,28562,1,538084,15267
%N A352032 Sum of the 4th powers of the odd proper divisors of n.
%H A352032 Seiichi Manyama, <a href="/A352032/b352032.txt">Table of n, a(n) for n = 1..10000</a>
%H A352032 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F A352032 a(n) = Sum_{d|n, d<n, d odd} d^4.
%F A352032 G.f.: Sum_{k>=1} (2*k-1)^4 * x^(4*k-2) / (1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Mar 02 2022
%F A352032 From _Amiram Eldar_, Oct 11 2023: (Start)
%F A352032 a(n) = A051001(n) - n^4*A000035(n).
%F A352032 Sum_{k=1..n} a(k) ~ c * n^5, where c = (zeta(5)-1)/10 = 0.0036927755... . (End)
%e A352032 a(10) = 626; a(10) = Sum_{d|10, d<10, d odd} d^4 = 1^4 + 5^4 = 626.
%t A352032 f[2, e_] := 1; f[p_, e_] := (p^(4*e+4) - 1)/(p^4 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^4, 0]; Array[a, 60] (* _Amiram Eldar_, Oct 11 2023 *)
%Y A352032 Sum of the k-th powers of the odd proper divisors of n for k=0..10: A091954 (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), this sequence (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).
%Y A352032 Cf. A000035, A013663, A051001.
%K A352032 nonn,easy
%O A352032 1,6
%A A352032 _Wesley Ivan Hurt_, Mar 01 2022