A351270 Sum of the 7th powers of the squarefree divisors of n.

1, 129, 2188, 129, 78126, 282252, 823544, 129, 2188, 10078254, 19487172, 282252, 62748518, 106237176, 170939688, 129, 410338674, 282252, 893871740, 10078254, 1801914272, 2513845188, 3404825448, 282252, 78126, 8094558822, 2188, 106237176, 17249876310, 22051219752, 27512614112

Offset

1,2

Comments

Inverse Möbius transform of n^7 * 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^7 * mu(d)^2.

Multiplicative with a(p^e) = 1 + p^7. - Amiram Eldar, Feb 06 2022

G.f.: Sum_{k>=1} mu(k)^2 * k^7 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022

Sum_{k=1..n} a(k) ~ c * n^8, where c = zeta(8)/(8*zeta(2)) = Pi^6/12600 = 0.0763007... . - Amiram Eldar, Nov 10 2022

Example

a(4) = 129; a(4) = Sum_{d|4} d^7 * mu(d)^2 = 1^7*1 + 2^7*1 + 4^7*0 = 129.

Mathematica

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^7); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

Crossrefs

Cf. A008683 (mu), A013661, A013666.

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), this sequence (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

Sequence in context: A230187 A240417 A353940 * A088719 A321563 A034681

Adjacent sequences: A351267 A351268 A351269 * A351271 A351272 A351273

Keyword

nonn,mult

Author

Wesley Ivan Hurt, Feb 05 2022

Status

approved