A199879 -id:A199879 - 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.

A199880 Engel expansion of 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

3, 4, 8, 12, 15, 33, 70, 4338, 22062, 46566, 98091, 255284, 2715877, 10855925, 150153128, 10009347774, 34679420772, 43644678207, 74587800101, 229110893125, 233558717156, 286861037311, 299617642336, 312870987050, 1632483095154, 31761226898013, 66327161231576

Offset: 1

Alois P. Heinz, Nov 11 2011

nonn

Cf. A006784 for definition of Engel expansion.

			0.42801103796472992390204...

F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191.

F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Eric Weisstein's World of Mathematics, Engel Expansion
Wikipedia, Engel Expansion
Index entries for sequences related to Engel expansions

Cf. A199814 (decimal expansion), A199879 (continued fraction).

f:= x-> (x^(x^x))^(x^x): g:= x-> x^(x^((x^x)^x)):
Digits:= 700:
xv:= fsolve(f(x)=g(x), x=0..0.99):
engel:= (r, n)-> `if`(n=0 or r=0, NULL, [ceil(1/r), engel(r*ceil(1/r)-1, n-1)][]):
engel(xv, 39);

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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