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%I #22 Oct 11 2023 03:54:39
%S 0,1,1,1,1,730,1,1,730,15626,1,730,1,117650,16355,1,1,532171,1,15626,
%T 118379,1771562,1,730,15626,4826810,532171,117650,1,11406980,1,1,
%U 1772291,24137570,133275,532171,1,47045882,4827539,15626,1,85884500,1,1771562,11938421,148035890
%N Sum of the 6th powers of the odd proper divisors of n.
%H Seiichi Manyama, <a href="/A352034/b352034.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F a(n) = Sum_{d|n, d<n, d odd} d^6.
%F G.f.: Sum_{k>=1} (2*k-1)^6 * x^(4*k-2) / (1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Mar 02 2022
%F For odd n >1, a(n) = A321810(n)-n^6; for even n, a(n) = A321810(n). - _R. J. Mathar_, Aug 15 2023
%F Sum_{k=1..n} a(k) ~ c * n^7, where c = (zeta(7)-1)/14 = 0.0005963769... . - _Amiram Eldar_, Oct 11 2023
%e a(10) = 15626; a(10) = Sum_{d|10, d<10, d odd} d^6 = 1^6 + 5^6 = 15626.
%t f[2, e_] := 1; f[p_, e_] := (p^(6*e+6) - 1)/(p^6 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^6, 0]; Array[a, 60] (* _Amiram Eldar_, Oct 11 2023 *)
%Y 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), A352032 (k=4), A352033 (k=5), this sequence (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).
%Y Cf. A013665, A321810.
%K nonn,easy
%O 1,6
%A _Wesley Ivan Hurt_, Mar 01 2022
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