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
a(n) = Sum_{d|n} lambda(d)*d^2*sigma_2(n/d), where lambda = A008836. - Ridouane Oudra, Jul 18 2025
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
(Python)
from math import prod
from sympy import factorint
def A351307(n): return prod((p**(4+((e&-2)<<1))-1)//(p**4-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jul 11 2024
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Wesley Ivan Hurt, Feb 06 2022
STATUS
approved