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

1,4

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

Links

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

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

Example

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

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 *)

Prog

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

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.

Sequence in context: A180699 A195529 A004217 * A051333 A273775 A182912

Adjacent sequences: A351306 A351307 A351308 * A351310 A351311 A351312

Keyword

nonn,easy,mult

Author

Wesley Ivan Hurt, Feb 06 2022

Status

approved