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A352035
Sum of the 7th powers of the odd proper divisors of n.
11
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
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).
Sequence in context: A079231 A017563 A357674 * A321811 A081865 A085442
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved