%I #32 Oct 22 2022 08:05:59

%S 1,2,3,5,5,6,7,11,10,10,11,15,13,14,15,23,17,20,19,25,21,22,23,33,26,

%T 26,31,35,29,30,31,47,33,34,35,50,37,38,39,55,41,42,43,55,50,46,47,69,

%U 50,52,51,65,53,62,55,77,57,58,59,75,61,62,70,95,65,66,67,85,69,70,71

%N a(n) = Sum_{ d divides n } phi(n)/phi(d).

%C a(n) = n iff n is squarefree number (cf. A005117).

%H Ivan Neretin, <a href="/A069208/b069208.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = (p^(e+1)-p^e+p^(e-1)-1)/(p-1).

%F a(n) = phi(n) * Sum_{k=1..n} 1/phi(n / gcd(n, k))^2. - _Daniel Suteu_, Nov 04 2018

%F a(n) = Sum_{k=1..n, gcd(n,k) = 1} tau(gcd(n,k-1)). - _Ilya Gutkovskiy_, Sep 24 2021

%F From _Werner Schulte_, Feb 27 2022: (Start)

%F Dirichlet convolution of A005361 and A000010.

%F Dirichlet convolution of A112526 and A000027.

%F Dirichlet g.f.: Sum_{n>0} a(n) / n^s = zeta(s-1) * zeta(2*s) * zeta(3*s) / zeta(6*s). (End)

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 15015/(2764*Pi^2) = 0.550411... . - _Amiram Eldar_, Oct 22 2022

%t Table[EulerPhi[n]*Total[1/EulerPhi@Divisors@n], {n, 71}] (* _Ivan Neretin_, Sep 20 2017 *)

%t f[p_, e_] := (p^(e + 1) - p^e + p^(e - 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Apr 14 2022 *)

%o (PARI) a(n) = sumdiv(n, d, eulerphi(n)/eulerphi(d)) \\ _Michel Marcus_, Jun 17 2013

%o (PARI) a(n) = my(f=factor(n)); prod(k=1, #f~, (f[k,1]^(f[k,2]-1) + (f[k,1]-1)*f[k,1]^f[k,2]-1) / (f[k,1]-1)); \\ _Daniel Suteu_, Nov 04 2018

%Y Cf. A000010, A000027, A005117, A005361, A069170, A112526.

%K nonn,mult

%O 1,2

%A _Vladeta Jovovic_, Apr 10 2002