A352036 Sum of the 8th powers of the odd proper divisors of n.

0, 1, 1, 1, 1, 6562, 1, 1, 6562, 390626, 1, 6562, 1, 5764802, 397187, 1, 1, 43053283, 1, 390626, 5771363, 214358882, 1, 6562, 390626, 815730722, 43053283, 5764802, 1, 2563287812, 1, 1, 214365443, 6975757442, 6155427, 43053283, 1, 16983563042, 815737283, 390626, 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^8.

G.f.: Sum_{k>=1} (2*k-1)^8 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A321812(n) - n^8*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^9, where c = (zeta(9)-1)/18 = 0.0001115773... . (End)

Example

a(10) = 390626; a(10) = Sum_{d|10, d<10, d odd} d^8 = 1^8 + 5^8 = 390626.

Mathematica

Table[Total[Select[Most[Divisors[n]], OddQ]^8], {n, 45}] (* Harvey P. Dale, Aug 07 2022 *)
f[2, e_] := 1; f[p_, e_] := (p^(8*e+8) - 1)/(p^8 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^8, 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), A352035 (k=7), this sequence (k=8), A352037 (k=9), A352038 (k=10).

Cf. A000035, A013667, A321812.

Sequence in context: A017312 A017432 A017564 * A321812 A031579 A288884

Adjacent sequences: A352033 A352034 A352035 * A352037 A352038 A352039

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 01 2022

Status

approved