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A352036
Sum of the 8th powers of the odd proper divisors of n.
11
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
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).
Sequence in context: A017312 A017432 A017564 * A321812 A031579 A288884
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved