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A344683 Dirichlet convolution of the Euler totient function with itself, applied twice.

%I A344683 #15 Jun 24 2022 11:50:42
%S A344683 1,3,6,9,12,18,18,25,30,36,30,54,36,54,72,66,48,90,54,108,108,90,66,
%T A344683 150,108,108,134,162,84,216,90,168,180,144,216,270,108,162,216,300,
%U A344683 120,324,126,270,360,198,138,396,234,324,288,324,156,402,360,450,324
%N A344683 Dirichlet convolution of the Euler totient function with itself, applied twice.
%C A344683 Dirichlet convolution of A000010 with A029935.
%H A344683 Sebastian Karlsson, <a href="/A344683/b344683.txt">Table of n, a(n) for n = 1..10000</a>
%F A344683 Dirichlet g.f.: zeta(s - 1)^3 / zeta(s)^3.
%F A344683 Multiplicative with a(p^e) = (1/2)*(p-1)*p^(e-3)*(e^2*(p-1)^2 + 3*e*(p^2-1) + 2*(p^2 + p + 1)).
%F A344683 Sum_{k=1..n} a(k) ~ 27*n^2/Pi^10 * (2*Pi^4*log(n)^2 - 2*Pi^4*log(n)*(1 + 6*log(2) - 72*(1/12 - zeta'(-1)) + 6*log(Pi)) + Pi^4*(1 + 6*gamma*(2*gamma - 1) - 12*sg1) + 864*zeta'(2)^2 - 36*Pi^2*((6*gamma - 1)*zeta'(2) + zeta''(2))), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - _Vaclav Kotesovec_, Jun 24 2022
%t A344683 f[p_, e_] := (1/2)*(p - 1)*p^(e - 3)*(e^2*(p - 1)^2 + 3*e*(p^2 - 1) + 2*(p^2 + p + 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Aug 17 2021 *)
%o A344683 (Python)
%o A344683 from sympy import divisors as div, totient as phi
%o A344683 def D(f, g, n):
%o A344683     return sum(f(d)*g(n//d) for d in div(n))
%o A344683 def phi_o_phi(n):
%o A344683     return D(phi, phi, n)
%o A344683 def a(n):
%o A344683     return D(phi, phi_o_phi, n)
%Y A344683 Cf. A000010, A029935, A166633.
%K A344683 nonn,mult
%O A344683 1,2
%A A344683 _Sebastian Karlsson_, Aug 17 2021