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a(n) = Sum_{d|n} phi(d) * J_2(n/d).
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%I #24 May 31 2024 04:13:18

%S 1,4,10,17,28,40,54,70,94,112,130,170,180,216,280,284,304,376,378,476,

%T 540,520,550,700,716,720,858,918,868,1120,990,1144,1300,1216,1512,

%U 1598,1404,1512,1800,1960,1720,2160,1890,2210,2632,2200,2254,2840,2682,2864,3040,3060,2860,3432,3640

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

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

%H Amiram Eldar, <a href="/A341772/b341772.txt">Table of n, a(n) for n = 1..10000</a>

%F Dirichlet g.f.: zeta(s-1) * zeta(s-2) / zeta(s)^2.

%F a(n) = Sum_{k=1..n} J_2(gcd(n,k)).

%F a(n) = Sum_{d|n} psi(d) * phi(d) * phi(n/d).

%F a(n) = Sum_{d|n} d * phi(d) * A029935(n/d).

%F a(n) = Sum_{d|n} d * sigma(d) * A007427(n/d).

%F a(n) = Sum_{d|n} d * A321322(n/d).

%F a(n) = Sum_{d|n} d * A023900(d) * A338164(n/d).

%F a(n) = Sum_{d|n} d^2 * A007431(n/d).

%F a(n) = Sum_{d|n} mu(n/d) * A069097(d).

%F Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (18*zeta(3)^2). - _Vaclav Kotesovec_, Feb 20 2021

%F 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

%F a(n) = Sum_{1 <= i, j <= n} phi(gcd(i, j, n)). - _Peter Bala_, Jan 21 2024

%F Multiplicative with a(p^e) = p^(e-3)*(p-1)*(p^e*(p+1)^2-p). - _Amiram Eldar_, May 31 2024

%t 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}]

%t 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 *)

%o (PARI) J2(n) = sumdiv(n, d, d^2 * moebius(n/d)); \\ A007434

%o a(n) = sumdiv(n, d, eulerphi(d) * J2(n/d)); \\ _Michel Marcus_, Feb 20 2021

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

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Feb 19 2021