OFFSET
1,2
COMMENTS
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Dirichlet g.f.: zeta(s-1) * zeta(s-2) / zeta(s)^2.
a(n) = Sum_{k=1..n} J_2(gcd(n,k)).
a(n) = Sum_{d|n} psi(d) * phi(d) * phi(n/d).
a(n) = Sum_{d|n} d * phi(d) * A029935(n/d).
a(n) = Sum_{d|n} d * sigma(d) * A007427(n/d).
a(n) = Sum_{d|n} d * A321322(n/d).
a(n) = Sum_{d|n} d^2 * A007431(n/d).
a(n) = Sum_{d|n} mu(n/d) * A069097(d).
Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (18*zeta(3)^2). - Vaclav Kotesovec, Feb 20 2021
a(n) = Sum_{k=1..n} J_2(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
a(n) = Sum_{1 <= i, j <= n} phi(gcd(i, j, n)). - Peter Bala, Jan 21 2024
Multiplicative with a(p^e) = p^(e-3)*(p-1)*(p^e*(p+1)^2-p). - Amiram Eldar, May 31 2024
MATHEMATICA
Jordan2[n_] := Sum[MoebiusMu[n/d] d^2, {d, Divisors[n]}]; a[n_] := Sum[EulerPhi[d] Jordan2[n/d], {d, Divisors[n]}]; Table[a[n], {n, 55}]
f[p_, e_] := p^(e-3)*(p-1)*(p^e*(p+1)^2-p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 31 2024 *)
PROG
(PARI) J2(n) = sumdiv(n, d, d^2 * moebius(n/d)); \\ A007434
a(n) = sumdiv(n, d, eulerphi(d) * J2(n/d)); \\ Michel Marcus, Feb 20 2021
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Ilya Gutkovskiy, Feb 19 2021
STATUS
approved