A351268 Sum of the 5th powers of the squarefree divisors of n.

1, 33, 244, 33, 3126, 8052, 16808, 33, 244, 103158, 161052, 8052, 371294, 554664, 762744, 33, 1419858, 8052, 2476100, 103158, 4101152, 5314716, 6436344, 8052, 3126, 12252702, 244, 554664, 20511150, 25170552, 28629152, 33, 39296688, 46855314, 52541808, 8052, 69343958

Offset

1,2

Comments

Inverse Möbius transform of n^5 * 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^5 * mu(d)^2.

Multiplicative with a(p^e) = 1 + p^5. - Amiram Eldar, Feb 06 2022

G.f.: Sum_{k>=1} mu(k)^2 * k^5 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022

Sum_{k=1..n} a(k) ~ c * n^6, where c = zeta(6)/(6*zeta(2)) = Pi^4/945 = 0.103078... . - Amiram Eldar, Nov 10 2022

Example

a(4) = 33; a(4) = Sum_{d|4} d^5 * mu(d)^2 = 1^5*1 + 2^5*1 + 4^4*0 = 33.

Mathematica

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^5); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

Crossrefs

Cf. A008683 (mu), A013661, A013664.

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), this sequence (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

Sequence in context: A274639 A306879 A178448 * A088703 A321561 A034679

Adjacent sequences: A351265 A351266 A351267 * A351269 A351270 A351271

Keyword

nonn,mult

Author

Wesley Ivan Hurt, Feb 05 2022

Status

approved