A352033 Sum of the 5th powers of the odd proper divisors of n.

0, 1, 1, 1, 1, 244, 1, 1, 244, 3126, 1, 244, 1, 16808, 3369, 1, 1, 59293, 1, 3126, 17051, 161052, 1, 244, 3126, 371294, 59293, 16808, 1, 762744, 1, 1, 161295, 1419858, 19933, 59293, 1, 2476100, 371537, 3126, 1, 4101152, 1, 161052, 821793, 6436344, 1, 244, 16808, 9768751

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^5.

G.f.: Sum_{k>=1} (2*k-1)^5 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A051002(n) - n^5*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^6, where c = (zeta(6)-1)/12 = 0.0014452551... . (End)

Example

a(10) = 3126; a(10) = Sum_{d|10, d<10, d odd} d^5 = 1^5 + 5^5 = 3126.

Mathematica

Table[Total[Select[Most[Divisors[n]], OddQ]^5], {n, 50}] (* Harvey P. Dale, May 01 2023 *)
f[2, e_] := 1; f[p_, e_] := (p^(5*e+5) - 1)/(p^5 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^5, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

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), A352031 (k=3), A352032 (k=4), this sequence (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Cf. A000035, A013664, A051002.

Sequence in context: A017561 A164657 A151638 * A248137 A243774 A051002

Adjacent sequences: A352030 A352031 A352032 * A352034 A352035 A352036

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 01 2022

Status

approved