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

Author

Wesley Ivan Hurt, Feb 05 2022

Keywords

• nonn
• mult

Comments

Inverse Möbius transform of n^3 * mu(n)^2. - Wesley Ivan Hurt, Jun 08 2023

Examples

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

Links

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

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).

Programs

Mathematica

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

PARI

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

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

Dirichlet g.f.: zeta(s)*zeta(s-3)/zeta(2s-6). - Michael Shamos, Feb 09 2025