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A352037
Sum of the 9th powers of the odd proper divisors of n.
11
0, 1, 1, 1, 1, 19684, 1, 1, 19684, 1953126, 1, 19684, 1, 40353608, 1972809, 1, 1, 387440173, 1, 1953126, 40373291, 2357947692, 1, 19684, 1953126, 10604499374, 387440173, 40353608, 1, 38445332184, 1, 1, 2357967375, 118587876498, 42306733, 387440173, 1, 322687697780
OFFSET
1,6
FORMULA
a(n) = Sum_{d|n, d<n, d odd} d^9.
G.f.: Sum_{k>=1} (2*k-1)^9 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022
From Amiram Eldar, Oct 11 2023: (Start)
a(n) = A321813(n) - n^9*A000035(n).
Sum_{k=1..n} a(k) ~ c * n^10, where c = (zeta(10)-1)/20 = 0.0000497287... . (End)
EXAMPLE
a(10) = 1953126; a(10) = Sum_{d|10, d<10, d odd} d^9 = 1^9 + 5^9 = 1953126.
MATHEMATICA
f[2, e_] := 1; f[p_, e_] := (p^(9*e+9) - 1)/(p^9 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^9, 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), A352036 (k=8), this sequence (k=9), A352038 (k=10).
Sequence in context: A017313 A017433 A017565 * A321813 A081866 A288885
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Mar 01 2022
STATUS
approved