A351309 Sum of the 4th powers of the square divisors of n.

1, 1, 1, 257, 1, 1, 1, 257, 6562, 1, 1, 257, 1, 1, 1, 65793, 1, 6562, 1, 257, 1, 1, 1, 257, 390626, 1, 6562, 257, 1, 1, 1, 65793, 1, 1, 1, 1686434, 1, 1, 1, 257, 1, 1, 1, 257, 6562, 1, 1, 65793, 5764802, 390626, 1, 257, 1, 6562, 1, 257, 1, 1, 1, 257, 1, 1, 6562, 16843009, 1

Offset: 1

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

• nonn
• easy
• mult

Comments

Inverse Möbius transform of n^4 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 29 2024

Examples

			a(16) = 65793; a(16) = Sum_{d^2|16} (d^2)^4 = (1^2)^4 + (2^2)^4 + (4^2)^4 = 65793.
		

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), A351308 (k=3), this sequence (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Cf. A010052, A008836, A001159.

Programs

Mathematica

f[p_, e_] := (p^(8*(1 + Floor[e/2])) - 1)/(p^8 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)
Table[Total[Select[Divisors[n],IntegerQ[Sqrt[#]]&]^4],{n,70}] (* Harvey P. Dale, Feb 11 2023 *)

PARI

a(n) = sumdiv(n, d, if (issquare(d), d^4)); \\ Michel Marcus, Jun 05 2024

Formula

a(n) = Sum_{d^2|n} (d^2)^4.

Multiplicative with a(p) = (p^(8*(1+floor(e/2))) - 1)/(p^8 - 1). - Amiram Eldar, Feb 07 2022

From Amiram Eldar, Sep 20 2023: (Start)

Dirichlet g.f.: zeta(s) * zeta(2*s-8).

Sum_{k=1..n} a(k) ~ (zeta(9/2)/9) * n^(9/2). (End)

G.f.: Sum_{k>=1} k^8 * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Jun 05 2024

a(n) = Sum_{d|n} d^4 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 29 2024

a(n) = Sum_{d|n} lambda(d)*d^4*sigma_4(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025