A352031 Sum of the cubes of the odd proper divisors of n.

0, 1, 1, 1, 1, 28, 1, 1, 28, 126, 1, 28, 1, 344, 153, 1, 1, 757, 1, 126, 371, 1332, 1, 28, 126, 2198, 757, 344, 1, 3528, 1, 1, 1359, 4914, 469, 757, 1, 6860, 2225, 126, 1, 9632, 1, 1332, 4257, 12168, 1, 28, 344, 15751, 4941, 2198, 1, 20440, 1457, 344, 6887, 24390, 1, 3528, 1

Offset: 1

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

• nonn
• easy

Examples

			a(10) = 126; a(10) = Sum_{d|10, d<10, d odd} d^3 = 1^3 + 5^3 = 126.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to sums of divisors.

Crossrefs

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).

Cf. A000035, A013662, A051000.

Programs

Mathematica

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 *)
Table[Total[Select[Most[Divisors[n]],OddQ]^3],{n,70}] (* Harvey P. Dale, Apr 14 2025 *)

PARI

a(n) = sumdiv(n/2^valuation(n,2), d, if ((dMichel Marcus, Mar 02 2022

Formula

a(n) = Sum_{d|n, d

G.f.: Sum_{k>=1} (2*k-1)^3 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A051000(n) - n^3*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(4)-1)/8 = 0.0102904042... . (End)

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