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%I #23 Oct 11 2023 03:54:12
%S 0,1,1,1,1,28,1,1,28,126,1,28,1,344,153,1,1,757,1,126,371,1332,1,28,
%T 126,2198,757,344,1,3528,1,1,1359,4914,469,757,1,6860,2225,126,1,9632,
%U 1,1332,4257,12168,1,28,344,15751,4941,2198,1,20440,1457,344,6887,24390,1,3528,1
%N Sum of the cubes of the odd proper divisors of n.
%H Seiichi Manyama, <a href="/A352031/b352031.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^3.
%F G.f.: Sum_{k>=1} (2*k-1)^3 * x^(4*k-2) / (1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Mar 02 2022
%F From _Amiram Eldar_, Oct 11 2023: (Start)
%F a(n) = A051000(n) - n^3*A000035(n).
%F Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(4)-1)/8 = 0.0102904042... . (End)
%e a(10) = 126; a(10) = Sum_{d|10, d<10, d odd} d^3 = 1^3 + 5^3 = 126.
%t f[2, e_] := 1; f[p_, e_] := (p^(3*e+3) - 1)/(p^3 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^3, 0]; Array[a, 60] (* _Amiram Eldar_, Oct 11 2023 *)
%o (PARI) a(n) = sumdiv(n/2^valuation(n,2), d, if ((d<n), d^3)); \\ _Michel Marcus_, Mar 02 2022
%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), this sequence (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).
%Y Cf. A000035, A013662, A051000.
%K nonn,easy
%O 1,6
%A _Wesley Ivan Hurt_, Mar 01 2022