A199814 Decimal expansion of x value of the unique pairwise intersection on (0,1) of distinct order 5 power tower functions with parentheses inserted.

4, 2, 8, 0, 1, 1, 0, 3, 7, 9, 6, 4, 7, 2, 9, 9, 2, 3, 9, 0, 2, 0, 4, 1, 6, 9, 3, 9, 1, 7, 5, 1, 2, 6, 5, 5, 3, 3, 7, 6, 7, 1, 0, 7, 3, 7, 8, 0, 3, 9, 3, 9, 2, 9, 2, 8, 5, 6, 7, 5, 4, 5, 9, 1, 3, 3, 3, 3, 9, 2, 4, 7, 5, 0, 2, 3, 3, 2, 9, 3, 1, 5, 9, 1, 0, 8, 1, 6, 7, 6, 4, 4, 2, 5, 0, 3, 0, 6, 7, 1, 9, 6, 5, 2, 4

Offset: 0

Author

Alois P. Heinz, Nov 10 2011

Keywords

• nonn
• cons

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)).

Examples

			0.42801103796472992390204...
		

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

Crossrefs

Cf. A000081 (number of distinct power tower functions), A000108 (number of parenthesizations), A199879 (continued fraction), A199880 (Engel expansion).

Programs

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 *)

Formula

x in (0,1) : x^(x^2)-2*x = 0.