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

1,6

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Index entries for sequences related to sums of divisors.

Formula

a(n) = Sum_{d|n, d<n, d odd} d^3.

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)

Example

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

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 *)

Prog

(PARI) a(n) = sumdiv(n/2^valuation(n, 2), d, if ((d<n), d^3)); \\ Michel Marcus, Mar 02 2022

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.

Sequence in context: A040784 A347162 A347174 * A174188 A176641 A040796

Adjacent sequences: A352028 A352029 A352030 * A352032 A352033 A352034

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 01 2022

Status

approved