A351311 Sum of the 6th powers of the square divisors of n.

1, 1, 1, 4097, 1, 1, 1, 4097, 531442, 1, 1, 4097, 1, 1, 1, 16781313, 1, 531442, 1, 4097, 1, 1, 1, 4097, 244140626, 1, 531442, 4097, 1, 1, 1, 16781313, 1, 1, 1, 2177317874, 1, 1, 1, 4097, 1, 1, 1, 4097, 531442, 1, 1, 16781313, 13841287202, 244140626, 1, 4097, 1, 531442, 1

Offset

1,4

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

a(n) = Sum_{d^2|n} (d^2)^6.

Multiplicative with a(p) = (p^(12*(1+floor(e/2))) - 1)/(p^12 - 1). - Amiram Eldar, Feb 07 2022

From Amiram Eldar, Sep 20 2023: (Start)

Dirichlet g.f.: zeta(s) * zeta(2*s-12).

Sum_{k=1..n} a(k) ~ (zeta(13/2)/13) * n^(13/2). (End)

Example

a(16) = 16781313; a(16) = Sum_{d^2|16} (d^2)^6 = (1^2)^6 + (2^2)^6 + (4^2)^6 = 16781313.

Mathematica

f[p_, e_] := (p^(12*(1 + Floor[e/2])) - 1)/(p^12 - 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), A351310 (k=5), this sequence (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Sequence in context: A017424 A017556 A309112 * A044887 A217196 A342685

Adjacent sequences: A351308 A351309 A351310 * A351312 A351313 A351314

Keyword

nonn,easy,mult

Author

Wesley Ivan Hurt, Feb 06 2022

Status

approved