This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A352031 #25 Apr 14 2025 17:39:11
%S A352031 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 A352031 126,2198,757,344,1,3528,1,1,1359,4914,469,757,1,6860,2225,126,1,9632,
%U A352031 1,1332,4257,12168,1,28,344,15751,4941,2198,1,20440,1457,344,6887,24390,1,3528,1
%N A352031 Sum of the cubes of the odd proper divisors of n.
%H A352031 Seiichi Manyama, <a href="/A352031/b352031.txt">Table of n, a(n) for n = 1..10000</a>
%H A352031 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F A352031 a(n) = Sum_{d|n, d<n, d odd} d^3.
%F A352031 G.f.: Sum_{k>=1} (2*k-1)^3 * x^(4*k-2) / (1 - x^(2*k-1)). - _Ilya Gutkovskiy_, Mar 02 2022
%F A352031 From _Amiram Eldar_, Oct 11 2023: (Start)
%F A352031 a(n) = A051000(n) - n^3*A000035(n).
%F A352031 Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(4)-1)/8 = 0.0102904042... . (End)
%e A352031 a(10) = 126; a(10) = Sum_{d|10, d<10, d odd} d^3 = 1^3 + 5^3 = 126.
%t A352031 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 *)
%t A352031 Table[Total[Select[Most[Divisors[n]],OddQ]^3],{n,70}] (* _Harvey P. Dale_, Apr 14 2025 *)
%o A352031 (PARI) a(n) = sumdiv(n/2^valuation(n,2), d, if ((d<n), d^3)); \\ _Michel Marcus_, Mar 02 2022
%Y A352031 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 A352031 Cf. A000035, A013662, A051000.
%K A352031 nonn,easy
%O A352031 1,6
%A A352031 _Wesley Ivan Hurt_, Mar 01 2022