OFFSET
1,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..1000
FORMULA
a(n) = Sum_{d|n} phi(n/d) * (n/d) * sigma_2(d^2)/sigma(d^2).
From Amiram Eldar, May 27 2024: (Start)
Multiplicative with a(p^e) = (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2.
Dirichlet g.f.: zeta(s) * zeta(s-2)^2 / zeta(s-1)^2.
Sum_{k=1..n} a(k) ~ (2*zeta(3)*n^3/(15*zeta(4))) * (log(n) + 2*gamma - 1/3 - 2*zeta'(2)/zeta(2) + zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620). (End)
MATHEMATICA
f[p_, e_] := (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 27 2024 *)
PROG
(PARI) a(n) = sum(i=1, n, sum(j=1, n, gcd(i, n)/gcd([i, j, n])));
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; (p^(2*e)*((e+1)*p^2 + 2*p-e) + 1)/(p+1)^2); } \\ Amiram Eldar, May 27 2024
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, May 21 2024
STATUS
approved