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

Author

Wesley Ivan Hurt, Feb 05 2022

Keywords

• nonn
• mult

Comments

Inverse Möbius transform of n^9 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

Examples

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane, Transforms.

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).

Programs

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 *)

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