A084220 - OEIS

%I #33 May 18 2024 11:35:07

%S 1,57,703,3641,15501,40071,117307,233017,512461,883557,1770231,

%T 2559623,4824613,6686499,10897203,14913081,24132657,29210277,47039023,

%U 56439141,82466821,100903167,148023723,163810951,242203001,275002941,373584043

%N a(n) = sigma_6(n^2)/sigma_3(n^2).

%H G. C. Greubel, <a href="/A084220/b084220.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = (p^(6*e + 3) + 1)/(p^3 + 1). - _Amiram Eldar_, Sep 13 2020

%F Sum_{k>=1} 1/a(k) = 1.019347996519986873084210965032965644185467985307512751244884310846924559959... - _Vaclav Kotesovec_, Sep 24 2020

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

%F From _Seiichi Manyama_, May 18 2024: (Start)

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

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

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

%t Table[DivisorSigma[6,n^2]/DivisorSigma[3,n^2],{n,30}] (* _Harvey P. Dale_, May 02 2012 *)

%t 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 *)

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

%o (PARI) a(n) = sigma(n^2, 6)/sigma(n^2, 3); \\ _Michel Marcus_, Oct 09 2018

%Y Cf. A001158, A013665, A013954.

%Y Cf. A057660, A084218.

%K nonn,mult

%O 1,2

%A _Benoit Cloitre_, Jun 21 2003