OFFSET
1,4
COMMENTS
Inverse Möbius transform of n^4 * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 29 2024
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{d^2|n} (d^2)^4.
Multiplicative with a(p) = (p^(8*(1+floor(e/2))) - 1)/(p^8 - 1). - Amiram Eldar, Feb 07 2022
From Amiram Eldar, Sep 20 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-8).
Sum_{k=1..n} a(k) ~ (zeta(9/2)/9) * n^(9/2). (End)
G.f.: Sum_{k>=1} k^8 * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Jun 05 2024
a(n) = Sum_{d|n} d^4 * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 29 2024
EXAMPLE
a(16) = 65793; a(16) = Sum_{d^2|16} (d^2)^4 = (1^2)^4 + (2^2)^4 + (4^2)^4 = 65793.
MATHEMATICA
f[p_, e_] := (p^(8*(1 + Floor[e/2])) - 1)/(p^8 - 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[#]]&]^4], {n, 70}] (* Harvey P. Dale, Feb 11 2023 *)
PROG
(PARI) a(n) = sumdiv(n, d, if (issquare(d), d^4)); \\ Michel Marcus, Jun 05 2024
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Wesley Ivan Hurt, Feb 06 2022
STATUS
approved