A199880 -id:A199880 - cp's OEIS frontend

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

Alois P. Heinz, Nov 10 2011

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)).

			0.42801103796472992390204...

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

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

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);

x /. FindRoot[x^(x^2) - 2*x == 0, {x, 1/2}, WorkingPrecision -> 110] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Dec 05 2013 *)

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

A199879 Continued fraction for x value of the unique pairwise intersection on (0,1) of distinct order 5 power tower functions with parentheses inserted.

Original entry on oeis.org

0, 2, 2, 1, 35, 1, 2, 2, 1, 2, 5, 3, 1, 1, 45, 1, 1, 6, 11, 2, 9, 2, 2, 2, 2, 1, 1, 1, 29, 1, 3, 7, 4, 1, 7, 61, 1, 1, 2, 1, 2, 6, 2, 1, 1, 96, 11, 1, 2, 1, 1, 4, 14, 1, 10, 1, 2, 1, 7, 4, 7, 5, 10, 1, 6, 2, 2, 9, 6, 8, 3, 1, 3, 1, 3, 7, 9

Offset: 0

Alois P. Heinz, Nov 11 2011

			0.42801103796472992390204...

Cf. A199814 (decimal expansion), A199880 (Engel expansion).

with(numtheory):
f:= x-> (x^(x^x))^(x^x): g:= x-> x^(x^((x^x)^x)):
Digits:= 200:
xv:= fsolve(f(x)=g(x), x=0..0.99):
cfrac(evalf(xv), 120, 'quotients')[];

terms = 77; digits = terms+10; xv = x /. FindRoot[x^(x^2) - 2x == 0, {x, 1/2}, WorkingPrecision -> digits]; ContinuedFraction[xv, terms] (* Jean-François Alcover, Mar 24 2017 *)

Offset changed by Andrew Howroyd, Jul 03 2024

cp's OEIS Frontend

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

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