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

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

• nonn
• easy
• mult

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

Examples

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

Links

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

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.

Programs

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 *)

PARI

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

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