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A144184 Decimal expansion of the convergent to the recurrence x = 1/(x^(1/x)-1/x-1) for all starting values of x >= 3.

%I A144184 #13 Feb 16 2025 08:33:09
%S A144184 5,5,0,7,9,8,5,6,5,2,7,7,3,1,7,8,2,5,7,5,8,9,0,2,6,2,9,8,0,5,2,1,3,8,
%T A144184 7,3,0,0,1,6,0,2,4,6,6,3,3,0,4,1,1,8,2,2,9,8,8,3,0,2,8,6,8,5,1,9,3,3,
%U A144184 6,8,2,3,8,2,0,3,9,0,2,5,8,1,7,5,5,8,0,6,6,4,8,9,4,9,7,9,6,3,9,4
%N A144184 Decimal expansion of the convergent to the recurrence x = 1/(x^(1/x)-1/x-1) for all starting values of x >= 3.
%C A144184 1/(x^(1/x)-1/x-1) ~ pi(x), the number of prime numbers <= x. This is comparable to the well known approximation Pi(x) ~ x/(log(x)-1). As x -> infinity, pi(x) - 1/(x^(1/x)-1/x-1) -> 1/2 as x-> infinity. This was derived from my original n-th root formula 1/(x^(1/x)-1) ~ pi(x). The convergent of the recurrence x = 1/(x^(1/x)-1) = 2.293166287... is expanded in A085846 and is referred to as Foias constant. The convergents 5.507985652... and 2.293166287... are both roots of 1/(x^(1/x)-1/x-1)-x = 0. 2.293166287... is also a root of 1/(x^(1/x)-1) - x = 0.
%C A144184 We have here examples of functions, f(x), for which we can solve for a root by recursion of the variable x. Another simple example is the recursion x = 1/(x+1).
%H A144184 Eric Weisstein, <a href="https://mathworld.wolfram.com/FoiasConstant.html">Foias Constant</a>
%F A144184 The convergent used to generate this sequence, 5.50798565277317825758902..., is computed with the recurrence x = 1/(x^(1/x)-1/x-1) and can also be found by solving for the roots of 1/(x^(1/x)-1/x-1)-x = 0.
%t A144184 RealDigits[ x /. FindRoot[ 1/(x^(1/x) - 1/x - 1) - x == 0, {x, 5}, WorkingPrecision -> 100]][[1]] (* _Jean-François Alcover_, Dec 20 2011 *)
%o A144184 (PARI) g(x) = 1/(x^(1/x)-1/x-1) g2(n) = a=n;for(j=1,100,a=g(a));b=eval(Vec(Str(floor(a*10^99))));
%o A144184 for(j=1,100,print1(b[j]","))
%Y A144184 Cf. A085846.
%K A144184 nonn,cons
%O A144184 1,1
%A A144184 _Cino Hilliard_, Sep 13 2008, Sep 15 2008