OFFSET
1,1
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
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.
MATHEMATICA
RealDigits[ x /. FindRoot[ 1/(x^(1/x) - 1/x - 1) - x == 0, {x, 5}, WorkingPrecision -> 100]][[1]] (* Jean-François Alcover, Dec 20 2011 *)
PROG
(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]", "))
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Cino Hilliard, Sep 13 2008, Sep 15 2008
STATUS
approved