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%I #6 Nov 26 2022 02:44:30
%S 1,3,0,7,3,2,1,3,7,1,7,0,6,0,7,2,3,6,9,2,9,6,4,2,2,8,0,4,2,5,3,9,8,8,
%T 3,9,1,4,2,7,4,3,4,6,8,6,0,8,2,3,9,4,0,9,8,0,1,5,3,6,3,5,6,9,8,1,7,0,
%U 0,9,7,0,8,9,0,0,8,4,9,7,3,2,2,0,0,7,2,0,2,5,4,0,4,5,4,8,4,4,8,1,2,9,7,2,9
%N Decimal expansion of the asymptotic mean of the e-unitary Euler function (A321167).
%H Nicusor Minculete and László Tóth, <a href="http://ac.inf.elte.hu/Vol_035_2011/elte_annales_35_jav_vagott.pdf#page=205">Exponential unitary divisors</a>, Annales Univ. Sci. Budapest., Sect. Comp. Vol. 35 (2011), pp. 205-216.
%F Equals lim_{m->oo} (1/m) Sum_{k=1..m} A321167(k).
%F Equals Product_{p prime} (1 + Sum_{e >= 3} (uphi(e) - uphi(e-1))/p^e), where uphi is the unitary totient function (A047994).
%e 1.307321371706072369296422804253988391427434686082394...
%t f[p_, e_] := p^e - 1; uphi[1] = 1; uphi[n_] := Times @@ f @@@ FactorInteger[n]; $MaxExtraPrecision = 500; m = 500; fun[x_] := Log[1 + Sum[x^e*(uphi[e] - uphi[e - 1]), {e, 3, m}]]; c = Rest[CoefficientList[Series[fun[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[Exp[fun[1/2] + NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]
%Y Cf. A047994, A321167, A327838.
%K nonn,cons
%O 1,2
%A _Amiram Eldar_, Nov 25 2022
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