A351308 Sum of the cubes of the square divisors of n.

1, 1, 1, 65, 1, 1, 1, 65, 730, 1, 1, 65, 1, 1, 1, 4161, 1, 730, 1, 65, 1, 1, 1, 65, 15626, 1, 730, 65, 1, 1, 1, 4161, 1, 1, 1, 47450, 1, 1, 1, 65, 1, 1, 1, 65, 730, 1, 1, 4161, 117650, 15626, 1, 65, 1, 730, 1, 65, 1, 1, 1, 65, 1, 1, 730, 266305, 1, 1, 1, 65, 1, 1, 1, 47450, 1

Offset

1,4

Comments

Inverse Möbius transform of n^3 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 20 2024

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d^2|n} (d^2)^3.

Multiplicative with a(p) = (p^(6*(1+floor(e/2))) - 1)/(p^6 - 1). - Amiram Eldar, Feb 07 2022

From Amiram Eldar, Sep 19 2023: (Start)

Dirichlet g.f.: zeta(s) * zeta(2*s-6).

Sum_{k=1..n} a(k) ~ (zeta(7/2)/7) * n^(7/2). (End)

G.f.: Sum_{k>=1} k^6 * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Jun 05 2024

a(n) = Sum_{d|n} d^3 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 20 2024

a(n) = Sum_{d|n} lambda(d)*d^3*sigma_3(n/d), where lambda = A008836. - Ridouane Oudra, Jul 18 2025

Example

a(16) = 4161; a(16) = Sum_{d^2|16} (d^2)^3 = (1^2)^3 + (2^2)^3 + (4^2)^3 = 4161.

Mathematica

f[p_, e_] := (p^(6*(1 + Floor[e/2])) - 1)/(p^6 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

Prog

(PARI) a(n) = sumdiv(n, d, if (issquare(d), d^3)); \\ Michel Marcus, Mar 24 2023

Crossrefs

Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), A035316 (k=1), A351307 (k=2), this sequence (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Cf. A010052, A261804 (zeta(7/2)), A008836, A001158.

Sequence in context: A378948 A204043 A295175 * A364072 A279290 A034061

Adjacent sequences: A351305 A351306 A351307 * A351309 A351310 A351311

Keyword

nonn,easy,mult

Author

Wesley Ivan Hurt, Feb 06 2022

Status

approved