A351314 Sum of the 8th powers of the square divisors of n.

1, 1, 1, 65537, 1, 1, 1, 65537, 43046722, 1, 1, 65537, 1, 1, 1, 4295032833, 1, 43046722, 1, 65537, 1, 1, 1, 65537, 152587890626, 1, 43046722, 65537, 1, 1, 1, 4295032833, 1, 1, 1, 2821153019714, 1, 1, 1, 65537, 1, 1, 1, 65537, 43046722, 1, 1, 4295032833, 33232930569602, 152587890626

Offset: 1

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

• nonn
• easy
• mult

Comments

Inverse Möbius transform of n^8 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

Examples

			a(16) = 4295032833; a(16) = Sum_{d^2|16} (d^2)^8 = (1^2)^8 + (2^2)^8 + (4^2)^8 = 4295032833.
		

Links

• Seiichi Manyama, 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), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), this sequence (k=8), A351315 (k=9), A351316 (k=10).

Cf. A010052, A008836, A013956.

Programs

Mathematica

f[p_, e_] := (p^(16*(1 + Floor[e/2])) - 1)/(p^16 - 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[#]]&]^8],{n,80}] (* Harvey P. Dale, Feb 13 2022 *)

PARI

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, k^16*x^k^2/(1-x^k^2))) \\ Seiichi Manyama, Feb 12 2022

Formula

a(n) = Sum_{d^2|n} (d^2)^8.

Multiplicative with a(p) = (p^(16*(1+floor(e/2))) - 1)/(p^16 - 1). - Amiram Eldar, Feb 07 2022

G.f.: Sum_{k>0} k^16*x^(k^2)/(1-x^(k^2)). - Seiichi Manyama, Feb 12 2022

From Amiram Eldar, Sep 20 2023: (Start)

Dirichlet g.f.: zeta(s) * zeta(2*s-16).

Sum_{k=1..n} a(k) ~ (zeta(17/2)/17) * n^(17/2). (End)

a(n) = Sum_{d|n} d^8 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024

a(n) = Sum_{d|n} lambda(d)*d^8*sigma_8(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025