A341772 a(n) = Sum_{d|n} phi(d) * J_2(n/d).

1, 4, 10, 17, 28, 40, 54, 70, 94, 112, 130, 170, 180, 216, 280, 284, 304, 376, 378, 476, 540, 520, 550, 700, 716, 720, 858, 918, 868, 1120, 990, 1144, 1300, 1216, 1512, 1598, 1404, 1512, 1800, 1960, 1720, 2160, 1890, 2210, 2632, 2200, 2254, 2840, 2682, 2864, 3040, 3060, 2860, 3432, 3640

Offset: 1

Author

Ilya Gutkovskiy, Feb 19 2021

Keywords

• nonn
• mult

Comments

Dirichlet convolution of Euler totient function phi (A000010) with Jordan function J_2 (A007434).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A000203, A001615, A002618, A007427, A007431, A007434, A008683, A023900, A029935, A064987, A069097, A321322, A322577, A338164.

Programs

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 *)

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

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 * A023900(d) * A338164(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