A351310 Sum of the 5th powers of the square divisors of n.

1, 1, 1, 1025, 1, 1, 1, 1025, 59050, 1, 1, 1025, 1, 1, 1, 1049601, 1, 59050, 1, 1025, 1, 1, 1, 1025, 9765626, 1, 59050, 1025, 1, 1, 1, 1049601, 1, 1, 1, 60526250, 1, 1, 1, 1025, 1, 1, 1, 1025, 59050, 1, 1, 1049601, 282475250, 9765626, 1, 1025, 1, 59050, 1, 1025, 1, 1

Offset

1,4

Comments

Inverse Möbius transform of n^5 * 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)^5.

Multiplicative with a(p) = (p^(10*(1+floor(e/2))) - 1)/(p^10 - 1). - Amiram Eldar, Feb 07 2022

From Amiram Eldar, Sep 20 2023: (Start)

Dirichlet g.f.: zeta(s) * zeta(2*s-10).

Sum_{k=1..n} a(k) ~ (zeta(11/2)/11) * n^(11/2). (End)

a(n) = Sum_{d|n} d^5 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 21 2024

a(n) = Sum_{d|n} lambda(d)*d^5*sigma_5(n/d), where lambda = A008836. - Ridouane Oudra, Jul 19 2025

Example

a(16) = 1049601; a(16) = Sum_{d^2|16} (d^2)^5 = (1^2)^5 + (2^2)^5 + (4^2)^5 = 1049601.

Mathematica

f[p_, e_] := (p^(10*(1 + Floor[e/2])) - 1)/(p^10 - 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), this sequence (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Cf. A010052, A008836, A001160.

Sequence in context: A031147 A087932 A395551 * A045035 A060948 A171385

Adjacent sequences: A351307 A351308 A351309 * A351311 A351312 A351313

Keyword

nonn,easy,mult

Author

Wesley Ivan Hurt, Feb 06 2022

Status

approved