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A029935 a(n) = Sum_{d divides n} phi(d)*phi(n/d).

%I A029935 #46 Jan 05 2024 00:01:31
%S A029935 1,2,4,5,8,8,12,12,16,16,20,20,24,24,32,28,32,32,36,40,48,40,44,48,56,
%T A029935 48,60,60,56,64,60,64,80,64,96,80,72,72,96,96,80,96,84,100,128,88,92,
%U A029935 112,120,112,128,120,104,120
%N A029935 a(n) = Sum_{d divides n} phi(d)*phi(n/d).
%C A029935 Dirichlet convolution of A000010 with itself. - _R. J. Mathar_, Aug 28 2015
%H A029935 Gheorghe Coserea, <a href="/A029935/b029935.txt">Table of n, a(n) for n = 1..20000</a>
%F A029935 From _Vladeta Jovovic_, Oct 30 2001: (Start)
%F A029935 Sum_{k=1..n} phi(gcd(n, k)).
%F A029935 Multiplicative with a(p^e) = (e+1)*(p^e - p^(e - 1)) - (e - 1)*(p^(e - 1) - p^(e - 2)). (End)
%F A029935 From _Franklin T. Adams-Watters_, Nov 19 2004: (Start)
%F A029935 Sum_{d|n} a(d) = A018804(n), Mobius transform of A018804.
%F A029935 Dirichlet g.f.: zeta(s-1)^2/zeta(s)^2. (End)
%F A029935 Equals row sums of triangle A143258. - _Gary W. Adamson_, Aug 02 2008
%F A029935 a(n) <= A000010(n) * A000005(n), with equality iff n = A005117(k) for some k. - _Gheorghe Coserea_, Oct 23 2016
%F A029935 Sum_{k=1..n} a(k) ~ 9*n^2 * ((2*log(n) + 4*gamma - 1)/Pi^4 - 24*Zeta'(2)/Pi^6), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jan 31 2019
%p A029935 with(numtheory): A029935 := proc(n) local i,j; j := 0; for i in divisors(n) do j := j+phi(i)*phi(n/i); od; j; end;
%t A029935 A029935[n_]:=DivisorSum[n,EulerPhi[#]*EulerPhi[n/#]&]; Array[A029935, 50]
%t A029935 f[p_, e_] := (e+1)*(p^e - p^(e-1)) - (e-1)*(p^(e-1) - p^(e-2)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Apr 26 2023 *)
%o A029935 (PARI)
%o A029935 a(n) = {
%o A029935   my(f = factor(n), fsz = matsize(f)[1],
%o A029935      g = prod(k=1, fsz, f[k,1]),
%o A029935      h = prod(k=1, fsz, sqr(f[k,1]-1)*f[k,2] + sqr(f[k,1])-1));
%o A029935   return(h*n\sqr(g));
%o A029935 };
%o A029935 vector(54, n, a(n))  \\ _Gheorghe Coserea_, Oct 23 2016
%o A029935 (PARI) a(n) = sumdiv(n, d, eulerphi(d)*eulerphi(n/d)); \\ _Michel Marcus_, Oct 23 2016
%Y A029935 Cf. A000005, A000010, A001620, A005117, A018804, A029936, A143258.
%Y A029935 Row sums of A159937.
%K A029935 mult,nonn
%O A029935 1,2
%A A029935 _N. J. A. Sloane_