A351313 Sum of the 7th powers of the square divisors of n.

1, 1, 1, 16385, 1, 1, 1, 16385, 4782970, 1, 1, 16385, 1, 1, 1, 268451841, 1, 4782970, 1, 16385, 1, 1, 1, 16385, 6103515626, 1, 4782970, 16385, 1, 1, 1, 268451841, 1, 1, 1, 78368963450, 1, 1, 1, 16385, 1, 1, 1, 16385, 4782970, 1, 1, 268451841, 678223072850, 6103515626, 1

Offset

1,4

Comments

Inverse Möbius transform of n^7 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

Links

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

Formula

a(n) = Sum_{d^2|n} (d^2)^7.

Multiplicative with a(p) = (p^(14*(1+floor(e/2))) - 1)/(p^14 - 1). - Amiram Eldar, Feb 07 2022

From Amiram Eldar, Sep 20 2023: (Start)

Dirichlet g.f.: zeta(s) * zeta(2*s-14).

Sum_{k=1..n} a(k) ~ (zeta(15/2)/15) * n^(15/2). (End)

a(n) = Sum_{d|n} d^7 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024

Example

a(16) = 268451841; a(16) = Sum_{d^2|16} (d^2)^7 = (1^2)^7 + (2^2)^7 + (4^2)^7 = 268451841.

Mathematica

f[p_, e_] := (p^(14*(1 + Floor[e/2])) - 1)/(p^14 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

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), A351309 (k=4), A351310 (k=5), A351311 (k=6), this sequence (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Cf. A010052.

Sequence in context: A223584 A185566 A160868 * A230635 A017691 A013962

Adjacent sequences: A351310 A351311 A351312 * A351314 A351315 A351316

Keyword

nonn,easy,mult

Author

Wesley Ivan Hurt, Feb 06 2022

Status

approved