A351273 Sum of the 10th powers of the squarefree divisors of n.

1, 1025, 59050, 1025, 9765626, 60526250, 282475250, 1025, 59050, 10009766650, 25937424602, 60526250, 137858491850, 289537131250, 576660215300, 1025, 2015993900450, 60526250, 6131066257802, 10009766650, 16680163512500, 26585860217050, 41426511213650, 60526250

Offset

1,2

Comments

Inverse Möbius transform of n^10 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

N. J. A. Sloane, Transforms.

Formula

a(n) = Sum_{d|n} d^10 * mu(d)^2.

Multiplicative with a(p^e) = 1 + p^10. - Amiram Eldar, Feb 06 2022

G.f.: Sum_{k>=1} mu(k)^2 * k^10 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022

Sum_{k=1..n} a(k) ~ c * n^11, where c = zeta(11)/(11*zeta(2)) = 0.0552934... . - Amiram Eldar, Nov 10 2022

Example

a(4) = 1025; a(4) = Sum_{d|4} d^10 * mu(d)^2 = 1^10*1 + 2^10*1 + 4^10*0 = 1025.

Mathematica

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^10); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)
Table[Total[Select[Divisors[n], SquareFreeQ]^10], {n, 25}] (* Harvey P. Dale, Nov 20 2022 *)

Crossrefs

Cf. A008683 (mu), A013661, A013669.

Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), this sequence (k=10).

Sequence in context: A171385 A031742 A353943 * A321807 A351305 A017683

Adjacent sequences: A351270 A351271 A351272 * A351274 A351275 A351276

Keyword

nonn,mult

Author

Wesley Ivan Hurt, Feb 05 2022

Status

approved