A351272 Sum of the 9th powers of the squarefree divisors of n.

1, 513, 19684, 513, 1953126, 10097892, 40353608, 513, 19684, 1001953638, 2357947692, 10097892, 10604499374, 20701400904, 38445332184, 513, 118587876498, 10097892, 322687697780, 1001953638, 794320419872, 1209627165996, 1801152661464, 10097892, 1953126, 5440108178862

Offset

1,2

Comments

Inverse Möbius transform of n^9 * 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^9 * mu(d)^2.

Multiplicative with a(p^e) = 1 + p^9. - Amiram Eldar, Feb 06 2022

G.f.: Sum_{k>=1} mu(k)^2 * k^9 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022

Sum_{k=1..n} a(k) ~ c * n^10, where c = zeta(10)/(10*zeta(2)) = Pi^8/155925 = 0.0608531... . - Amiram Eldar, Nov 10 2022

Example

a(4) = 513; a(4) = Sum_{d|4} d^9 * mu(d)^2 = 1^9*1 + 2^9*1 + 4^9*0 = 513.

Mathematica

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^9); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)
Table[Total[Select[Divisors[n], SquareFreeQ]^9], {n, 30}] (* Harvey P. Dale, Feb 21 2023 *)

Crossrefs

Cf. A008683 (mu), A013661, A013668.

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), this sequence (k=9), A351273 (k=10).

Sequence in context: A230188 A223651 A353942 * A321565 A351304 A017681

Adjacent sequences: A351269 A351270 A351271 * A351273 A351274 A351275

Keyword

nonn,mult

Author

Wesley Ivan Hurt, Feb 05 2022

Status

approved