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

1,4

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

Links

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

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

Example

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

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 *)

Prog

(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

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.

Sequence in context: A017446 A017578 A370255 * A017703 A013968 A036098

Adjacent sequences: A351313 A351314 A351315 * A351317 A351318 A351319

Keyword

nonn,easy,mult

Author

Wesley Ivan Hurt, Feb 06 2022

Status

approved