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

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, 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, 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

Offset: 1

Author

Cino Hilliard, Sep 13 2008, Sep 15 2008

Keywords

• nonn
• cons

Comments

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.

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).

Links

• Eric Weisstein, Foias Constant

Crossrefs

Cf. A085846.

Programs

Mathematica

RealDigits[ x /. FindRoot[ 1/(x^(1/x) - 1/x - 1) - x == 0, {x, 5}, WorkingPrecision -> 100]][[1]] (* Jean-François Alcover, Dec 20 2011 *)

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))));
for(j=1,100,print1(b[j]","))

Formula

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.