%I #8 Feb 28 2023 08:02:23

%S 1,2,5,2,7,0,7,7,8,5,3,7,5,4,4,6,1,2,6,0,5,3,7,5,0,7,5,1,9,3,4,2,8,3,

%T 0,6,0,4,3,9,2,3,7,9,6,7,1,0,8,9,1,5,3,7,3,7,4,4,8,4,9,5,1,4,0,2,9,5,

%U 7,8,3,4,3,8,6,5,4,4,2,8,6,5,0,9,5,3,7

%N Decimal expansion of the asymptotic mean of the exponential totient function (A072911).

%H László Tóth, <a href="http://ac.inf.elte.hu/Vol_024_2004/285.pdf">On certain arithmetic functions involving exponential divisors</a>, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-294.

%F Equals lim_{m->oo} (1/m) Sum_{k=1..m} A072911(k).

%F Equals Product_{p prime} (1 + Sum_{e >= 3} (phi(e) - phi(e-1))/p^e), where phi is the Euler totient function (A000010).

%e 1.252707785375446126053750751934283060439237967108915...

%t $MaxExtraPrecision = 500; m = 500; f[x_] := Log[1 + Sum[x^e * (EulerPhi[e] - EulerPhi[e - 1]), {e, 3, m}]]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[Exp[f[1/2] + NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

%Y Cf. A000010, A072911, A322887, A327837, A361013.

%K nonn,cons

%O 1,2

%A _Amiram Eldar_, Sep 27 2019