A351269 Sum of the 6th powers of the squarefree divisors of n.

1, 65, 730, 65, 15626, 47450, 117650, 65, 730, 1015690, 1771562, 47450, 4826810, 7647250, 11406980, 65, 24137570, 47450, 47045882, 1015690, 85884500, 115151530, 148035890, 47450, 15626, 313742650, 730, 7647250, 594823322, 741453700, 887503682, 65, 1293240260

Offset

1,2

Comments

Inverse Möbius transform of n^6 * 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^6 * mu(d)^2.

Multiplicative with a(p^e) = 1 + p^6. - Amiram Eldar, Feb 06 2022

G.f.: Sum_{k>=1} mu(k)^2 * k^6 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022

Sum_{k=1..n} a(k) ~ c * n^7, where c = zeta(7)/(7*zeta(2)) = 0.0875718... . - Amiram Eldar, Nov 10 2022

Example

a(4) = 65; a(4) = Sum_{d|4} d^6 * mu(d)^2 = 1^6*1 + 2^6*1 + 4^6*0 = 65.

Mathematica

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^6); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

Crossrefs

Cf. A008683 (mu), A013661, A013665.

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

Sequence in context: A200890 A268265 A353939 * A088677 A321562 A034680

Adjacent sequences: A351266 A351267 A351268 * A351270 A351271 A351272

Keyword

nonn,mult

Author

Wesley Ivan Hurt, Feb 05 2022

Status

approved