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

Author

Wesley Ivan Hurt, Mar 01 2022

Keywords

• nonn
• easy

Examples

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

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), A352034 (k=6), A352035 (k=7), this sequence (k=8), A352037 (k=9), A352038 (k=10).

Cf. A000035, A013667, A321812.

Programs

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 *)

Formula

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

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)

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