A352034 Sum of the 6th powers of the odd proper divisors of n.

0, 1, 1, 1, 1, 730, 1, 1, 730, 15626, 1, 730, 1, 117650, 16355, 1, 1, 532171, 1, 15626, 118379, 1771562, 1, 730, 15626, 4826810, 532171, 117650, 1, 11406980, 1, 1, 1772291, 24137570, 133275, 532171, 1, 47045882, 4827539, 15626, 1, 85884500, 1, 1771562, 11938421, 148035890

Offset: 1

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

• nonn
• easy

Examples

			a(10) = 15626; a(10) = Sum_{d|10, d<10, d odd} d^6 = 1^6 + 5^6 = 15626.
		

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), A352033 (k=5), this sequence (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Cf. A013665, A321810.

Programs

Mathematica

f[2, e_] := 1; f[p_, e_] := (p^(6*e+6) - 1)/(p^6 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^6, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)
Table[Total[Select[Most[Divisors[n]],OddQ]^6],{n,50}] (* Harvey P. Dale, Sep 15 2024 *)

Formula

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

G.f.: Sum_{k>=1} (2*k-1)^6 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

For odd n >1, a(n) = A321810(n)-n^6; for even n, a(n) = A321810(n). - R. J. Mathar, Aug 15 2023

Sum_{k=1..n} a(k) ~ c * n^7, where c = (zeta(7)-1)/14 = 0.0005963769... . - Amiram Eldar, Oct 11 2023

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