OFFSET
0,1
COMMENTS
Order 5 is the smallest order such that pairwise intersections on (0,1) of distinct power tower functions with parentheses inserted exist. The corresponding y value is 0.66337467860163682654502... . The two intersecting functions are x-> (x^(x^x))^(x^x) and x-> x^(x^((x^x)^x)).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vladimir Reshetnikov, Intersections of x^x^...^x, SeqFan Discussion, Nov 2011.
Eric Weisstein's World of Mathematics, Power Tower
FORMULA
x in (0,1) : x^(x^2)-2*x = 0.
EXAMPLE
0.42801103796472992390204...
MAPLE
f:= x-> (x^(x^x))^(x^x): g:= x-> x^(x^((x^x)^x)):
nmax:= 140: Digits:= nmax+10:
xv:= fsolve(f(x)=g(x), x=0..0.99):
s:= convert(xv, string):
seq(parse(s[n+2]), n=0..nmax);
MATHEMATICA
x /. FindRoot[x^(x^2) - 2*x == 0, {x, 1/2}, WorkingPrecision -> 110] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Dec 05 2013 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Alois P. Heinz, Nov 10 2011
STATUS
approved