A352038 Sum of the 10th powers of the odd proper divisors of n.

0, 1, 1, 1, 1, 59050, 1, 1, 59050, 9765626, 1, 59050, 1, 282475250, 9824675, 1, 1, 3486843451, 1, 9765626, 282534299, 25937424602, 1, 59050, 9765626, 137858491850, 3486843451, 282475250, 1, 576660215300, 1, 1, 25937483651, 2015993900450, 292240875, 3486843451

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^10.

G.f.: Sum_{k>=1} (2*k-1)^10 * x^(4*k-2) / (1 - x^(2*k-1)). - Ilya Gutkovskiy, Mar 02 2022

From Amiram Eldar, Oct 11 2023: (Start)

a(n) = A321814(n) - n^10*A000035(n).

Sum_{k=1..n} a(k) ~ c * n^11, where c = (zeta(11)-1)/22 = 0.0000224631... . (End)

Example

a(10) = 9765626; a(10) = Sum_{d|10, d<10, d odd} d^10 = 1^10 + 5^10 = 9765626.

Mathematica

f[2, e_] := 1; f[p_, e_] := (p^(10*e+10) - 1)/(p^10 - 1); a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - If[OddQ[n], n^10, 0]; Array[a, 60] (* Amiram Eldar, Oct 11 2023 *)

Prog

(Python)
from math import prod
from sympy import factorint
def A352038(n): return prod((p**(10*(e+1))-1)//(p**10-1) for p, e in factorint(n).items() if p > 2) - (n**10 if n % 2 else 0) # Chai Wah Wu, Mar 01 2022

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), A352037 (k=9), this sequence (k=10).

Cf. A000035, A013669, A321814.

Sequence in context: A017314 A017434 A017566 * A321814 A288886 A210172

Adjacent sequences: A352035 A352036 A352037 * A352039 A352040 A352041

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Mar 01 2022

Status

approved