A351316 Sum of the 10th powers of the square divisors of n.

1, 1, 1, 1048577, 1, 1, 1, 1048577, 3486784402, 1, 1, 1048577, 1, 1, 1, 1099512676353, 1, 3486784402, 1, 1048577, 1, 1, 1, 1048577, 95367431640626, 1, 3486784402, 1048577, 1, 1, 1, 1099512676353, 1, 1, 1, 3656161927895954, 1, 1, 1, 1048577, 1, 1, 1, 1048577, 3486784402, 1, 1

Offset: 1

Author

Wesley Ivan Hurt, Feb 06 2022

Keywords

• nonn
• easy
• mult

Comments

Inverse Möbius transform of n^10 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 21 2024

Examples

			a(16) = 1099512676353; a(16) = Sum_{d^2|16} (d^2)^10 = (1^2)^10 + (2^2)^10 + (4^2)^10 = 1099512676353.
		

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), A351314 (k=8), A351315 (k=9), this sequence (k=10).

Cf. A010052, A008836, A013958.

Programs

Mathematica

f[p_, e_] := (p^(20*(1 + Floor[e/2])) - 1)/(p^20 - 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[#]]&]^10],{n,50}] (* Harvey P. Dale, Aug 24 2024 *)

PARI

my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, k^20*x^k^2/(1-x^k^2))) \\ Seiichi Manyama, Feb 12 2022

Formula

a(n) = Sum_{d^2|n} (d^2)^10.

Multiplicative with a(p) = (p^(20*(1+floor(e/2))) - 1)/(p^20 - 1). - Amiram Eldar, Feb 07 2022

G.f.: Sum_{k>0} k^20*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-20).

Sum_{k=1..n} a(k) ~ (zeta(21/2)/21) * n^(21/2). (End)

a(n) = Sum_{d|n} d^10 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024

a(n) = Sum_{d|n} lambda(d)*d^10*sigma_10(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025