A352035 Sum of the 7th powers of the odd proper divisors of n.

0, 1, 1, 1, 1, 2188, 1, 1, 2188, 78126, 1, 2188, 1, 823544, 80313, 1, 1, 4785157, 1, 78126, 825731, 19487172, 1, 2188, 78126, 62748518, 4785157, 823544, 1, 170939688, 1, 1, 19489359, 410338674, 901669, 4785157, 1, 893871740, 62750705, 78126, 1, 1801914272, 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^7.

G.f.: Sum_{k>=1} (2*k-1)^7 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A321811(n) - n^7*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^8, where c = (zeta(8)-1)/16 = 0.0002548347... . (End)

Example

a(10) = 78126; a(10) = Sum_{d|10, d<10, d odd} d^7 = 1^7 + 5^7 = 78126.

Mathematica

f[2, e_] := 1; f[p_, e_] := (p^(7*e+7) - 1)/(p^7 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^7, 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), A352033 (k=5), A352034 (k=6), this sequence (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Cf. A000035, A013666, A321811.

Sequence in context: A079231 A017563 A357674 * A321811 A081865 A085442

Adjacent sequences: A352032 A352033 A352034 * A352036 A352037 A352038

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 01 2022

Status

approved