A351271 Sum of the 8th powers of the squarefree divisors of n.

1, 257, 6562, 257, 390626, 1686434, 5764802, 257, 6562, 100390882, 214358882, 1686434, 815730722, 1481554114, 2563287812, 257, 6975757442, 1686434, 16983563042, 100390882, 37828630724, 55090232674, 78310985282, 1686434, 390626, 209642795554, 6562, 1481554114, 500246412962

Offset

1,2

Comments

Inverse Möbius transform of n^8 * 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^8 * mu(d)^2.

Multiplicative with a(p^e) = 1 + p^8. - Amiram Eldar, Feb 06 2022

G.f.: Sum_{k>=1} mu(k)^2 * k^8 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022

Sum_{k=1..n} a(k) ~ c * n^9, where c = zeta(9)/(9*zeta(2)) = 0.0676831... . - Amiram Eldar, Nov 10 2022

Example

a(4) = 257; a(4) = Sum_{d|4} d^8 * mu(d)^2 = 1^8*1 + 2^8*1 + 4^8*0 = 257.

Mathematica

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^8); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

Crossrefs

Cf. A008683 (mu), A013661, A013667.

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), this sequence (k=8), A351272 (k=9), A351273 (k=10).

Sequence in context: A209533 A125648 A353941 * A155468 A321564 A034682

Adjacent sequences: A351268 A351269 A351270 * A351272 A351273 A351274

Keyword

nonn,mult

Author

Wesley Ivan Hurt, Feb 05 2022

Status

approved