A352037 Sum of the 9th powers of the odd proper divisors of n.

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

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^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).

Cf. A000035, A013668, A321813.

Sequence in context: A017313 A017433 A017565 * A321813 A081866 A288885

Adjacent sequences: A352034 A352035 A352036 * A352038 A352039 A352040

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 01 2022

Status

approved