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%I #10 May 14 2019 11:04:26
%S 8,1,9,1,9,0,9,6,4,1,4,8,9,9,1,9,0,8,1,8,0,3,6,5,6,6,0,3,8,1,3,7,3,5,
%T 8,2,7,2,2,2,6,8,8,5,2,4,7,9,7,1,8,4,9,8,5,8,2,1,4,6,6,0,3,7,6,2,1,1,
%U 7,4,3,5,0,4,7,2,2,0,4,0,2,2,0,8,7,0,7
%N Decimal expansion of the asymptotic mean of 2^omega(k)/d(k), where omega(k) is the number of distinct prime divisors of k (A001221) and d(k) is its number of divisors (A000005).
%C Also the asymptotic mean of the ratio between the number of unitary divisors and the number of divisors of the integers.
%F Equals Product_{p prime} (1-1/p)*(2*p*log(p/(p-1))-1).
%e 0.81919096414899190818036566038137358272226885247971...
%t $MaxExtraPrecision = 1000; m = 1000; f[p_] := (1 - 1/p)*(2*p*Log[p/(p - 1)] - 1); c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; RealDigits[f[2] * Exp[ NSum[ Indexed[c, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]
%Y Cf. A000005, A001221, A034444, A307870 (mean of the inverse ratio).
%K nonn,cons
%O 0,1
%A _Amiram Eldar_, May 10 2019
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