OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = Sum_{1 <= x_1, x_2, x_3, x_4, x_5 <= n} ( gcd(x_1, n)/gcd(x_1, x_2, x_3, x_4, x_5, n) )^5.
a(n) = Sum_{d|n} mu(n/d) * (n/d)^4 * sigma_9(d).
a(n) = Sum_{d|n} phi(n/d) * (n/d)^8 * sigma_8(d^2)/sigma_4(d^2).
From Amiram Eldar, May 25 2024: (Start)
Multiplicative with a(p^e) = (p^(9*e+9) - p^(9*e+4) + p^4 - 1)/(p^9-1).
Dirichlet g.f.: zeta(s)*zeta(s-9)/zeta(s-4).
Sum_{k=1..n} a(k) ~ c * n^10 / 10, where c = zeta(10)/zeta(6) = Pi^4/99 = 0.983930212464... . (End)
MATHEMATICA
f[p_, e_] := (p^(9*e+9) - p^(9*e+4) + p^4 - 1)/(p^9-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 24] (* Amiram Eldar, May 25 2024 *)
PROG
(PARI) a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^4*sigma(d, 9));
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^8*sigma(d^2, 8)/sigma(d^2, 4));
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 25 2024
STATUS
approved