A199814 -id:A199814 - cp's OEIS frontend

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

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

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

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

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