OFFSET
1,1
COMMENTS
The value of x where f(x) is maximum is 8.06157... Note that greater precision in this value is made difficult due to a broad "flat" maximum.
In the recursive formula: b(n+1) = log(b(n))^log(b(n)) + c, where c is a constant, the maximum value of c, without the recursion diverging to infinity, is maximum value of the function above (3.4172192....), with b(1) set anywhere in the range 1 < b(1) <= 8.06157.
At values of c between 3.4172192 +/- 0.0000005 demonstrable convergence or divergence of the recursion above takes tens of thousands of iterations, increasing with further closeness to 3.4172192517331..., if b(1) is within the range above.
At x = e^e = 15.1542... (see A073226), the function f(x) = 0. This is the only value for b(1) where the recursion above is stable when c = 0.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
EXAMPLE
3.4172192517331903782941430626511991141665169728869621034583784206062...
MATHEMATICA
digits = 92; FindMaximum[x-Log[x]^Log[x], {x, 3}, WorkingPrecision -> digits] // First // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 20 2014 *)
PROG
(PARI) (x->x-log(x)^log(x))(solve(x=8, 9, my(L=log(x)); 1-L^L*(1+log(L))/x)) \\ Charles R Greathouse IV, Jun 18 2013
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Richard R. Forberg, Jun 17 2013
STATUS
approved