|
|
A084220
|
|
a(n) = sigma_6(n^2)/sigma_3(n^2).
|
|
10
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
LINKS
|
|
|
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
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)
|
|
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 *)
|
|
PROG
|
(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
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,mult
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|