A084220 a(n) = sigma_6(n^2)/sigma_3(n^2).

1, 57, 703, 3641, 15501, 40071, 117307, 233017, 512461, 883557, 1770231, 2559623, 4824613, 6686499, 10897203, 14913081, 24132657, 29210277, 47039023, 56439141, 82466821, 100903167, 148023723, 163810951, 242203001, 275002941, 373584043

Offset: 1

Author

Benoit Cloitre, Jun 21 2003

Keywords

• nonn
• mult

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001158, A013665, A013954.

Cf. A057660, A084218.

Programs

Maple

with(numtheory): a:=n->sigma[6](n^2)/sigma[3](n^2): seq(a(n),n=1..30); # Muniru A Asiru, Oct 09 2018

Mathematica

Table[DivisorSigma[6,n^2]/DivisorSigma[3,n^2],{n,30}] (* Harvey P. Dale, May 02 2012 *)
f[p_, e_] := (p^(6*e + 3) + 1)/(p^3 + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 30] (* Amiram Eldar, Sep 13 2020 *)

PARI

a(n)=sumdiv(n^2,d,d^6)/sumdiv(n^2,d,d^3)

PARI

a(n) = sigma(n^2, 6)/sigma(n^2, 3); \\ Michel Marcus, Oct 09 2018

Formula

Multiplicative with a(p^e) = (p^(6*e + 3) + 1)/(p^3 + 1). - Amiram Eldar, Sep 13 2020

Sum_{k>=1} 1/a(k) = 1.019347996519986873084210965032965644185467985307512751244884310846924559959... - Vaclav Kotesovec, Sep 24 2020

Sum_{k=1..n} a(k) ~ c * n^7, where c = 90*zeta(7)/(7*Pi^4) = 0.133093... . - Amiram Eldar, Oct 30 2022

From Seiichi Manyama, May 18 2024: (Start)

a(n) = Sum_{1 <= x_1, x_2, x_3 <= n} ( n/gcd(x_1, x_2, x_3, n) )^3.

a(n) = Sum_{d|n} mu(n/d) * (n/d)^3 * sigma_6(d). (End)