A351307 Sum of the squares of the square divisors of n.

1, 1, 1, 17, 1, 1, 1, 17, 82, 1, 1, 17, 1, 1, 1, 273, 1, 82, 1, 17, 1, 1, 1, 17, 626, 1, 82, 17, 1, 1, 1, 273, 1, 1, 1, 1394, 1, 1, 1, 17, 1, 1, 1, 17, 82, 1, 1, 273, 2402, 626, 1, 17, 1, 82, 1, 17, 1, 1, 1, 17, 1, 1, 82, 4369, 1, 1, 1, 17, 1, 1, 1, 1394, 1, 1, 626, 17, 1, 1

Offset

1,4

Comments

Inverse Möbius transform of n^2 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 20 2024

Links

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

Formula

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

Multiplicative with a(p) = (p^(4*(1+floor(e/2))) - 1)/(p^4 - 1). - Amiram Eldar, Feb 07 2022

G.f.: Sum_{k>0} k^4*x^(k^2)/(1-x^(k^2)). - Seiichi Manyama, Feb 12 2022

From Amiram Eldar, Sep 19 2023: (Start)

Dirichlet g.f.: zeta(s) * zeta(2*s-4).

Sum_{k=1..n} a(k) ~ (zeta(5/2)/5) * n^(5/2). (End)

a(n) = Sum_{d|n} d^2 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 20 2024

Example

a(16) = 273; a(16) = Sum_{d^2|16} (d^2)^2 = (1^2)^2 + (2^2)^2 + (4^2)^2 = 273.

Mathematica

f[p_, e_] := (p^(4*(1 + Floor[e/2])) - 1)/(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 07 2022 *)

Prog

(PARI) my(N=99, x='x+O('x^N)); Vec(sum(k=1, N, k^4*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), this sequence (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).

Cf. A010052, A247041 (zeta(5/2)).

Sequence in context: A089170 A040292 A040293 * A040294 A040291 A040290

Adjacent sequences: A351304 A351305 A351306 * A351308 A351309 A351310

Keyword

nonn,easy,mult

Author

Wesley Ivan Hurt, Feb 06 2022

Status

approved