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

1,4

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

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

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

Example

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

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 *)

Prog

(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

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.

Sequence in context: A013881 A027747 A255323 * A051332 A123388 A070816

Adjacent sequences: A351311 A351312 A351313 * A351315 A351316 A351317

Keyword

nonn,easy,mult

Author

Wesley Ivan Hurt, Feb 06 2022

Status

approved