A351266 Sum of the cubes of the squarefree divisors of n.

1, 9, 28, 9, 126, 252, 344, 9, 28, 1134, 1332, 252, 2198, 3096, 3528, 9, 4914, 252, 6860, 1134, 9632, 11988, 12168, 252, 126, 19782, 28, 3096, 24390, 31752, 29792, 9, 37296, 44226, 43344, 252, 50654, 61740, 61544, 1134, 68922, 86688, 79508, 11988, 3528, 109512, 103824

Offset

1,2

Comments

Inverse Möbius transform of n^3 * 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^3 * mu(d)^2.

a(n) = abs(A328640(n)).

G.f.: Sum_{k>=1} mu(k)^2 * k^3 * x^k / (1 - x^k). - Ilya Gutkovskiy, Feb 06 2022

Multiplicative with a(p^e) = 1 + p^3. - Amiram Eldar, Feb 06 2022

Sum_{k=1..n} a(k) ~ c * n^4, where c = zeta(4)/(4*zeta(2)) = Pi^2/60 = 0.164493... . - Amiram Eldar, Nov 10 2022

Example

a(4) = 9; a(4) = Sum_{d|4} d^3 * mu(d)^2 = 1^3*1 + 2^3*1 + 4^3*0 = 9.

Mathematica

a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^3); Array[a, 100] (* Amiram Eldar, Feb 06 2022 *)

Prog

(PARI) a(n) = sumdiv(n, d, if (issquarefree(d), d^3)); \\ Michel Marcus, Feb 06 2022

Crossrefs

Cf. A008683 (mu), A013661, A013662, A328640.

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), this sequence (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).

Sequence in context: A225300 A053825 A328640 * A369721 A369759 A033479

Adjacent sequences: A351263 A351264 A351265 * A351267 A351268 A351269

Keyword

nonn,mult

Author

Wesley Ivan Hurt, Feb 05 2022

Status

approved