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

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

• nonn
• easy

Examples

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

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), 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.

Programs

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 *)

Formula

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

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)

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.