A225134 -id:A225134 - cp's OEIS frontend

A225153 Continued fraction for the positive root of x^x^x^x = 2 (A225134).

1, 2, 4, 5, 1, 1, 184, 1, 1, 8, 1, 7, 1, 12, 3, 1, 4, 2, 1, 2, 1, 125, 1, 2, 1, 1, 2, 2, 5, 12, 7, 1, 8, 2, 1, 6, 1, 3, 2, 1, 2, 1, 14, 1, 1, 1, 3, 1, 1, 6485, 1, 1, 1, 3, 1, 2, 1, 1, 1, 17, 1, 2, 3, 3, 3, 2, 7, 1, 2, 1, 8, 1, 9, 1, 1, 7, 1, 4, 9, 1, 1, 1, 1, 3, 2

Offset: 0

Vladimir Reshetnikov, Apr 30 2013

x = 1.44660143242986417... = 1 + 1/(2 + 1/(4 + 1/(5 + 1/(1 + 1/(1 + 1/(184 + 1/(...))))))).
This constant is sometimes called the 4th super-root of 2.
It is unknown if it is rational, algebraic irrational, or transcendental. Hence, it is unknown if this continued fraction is aperiodic, or even if it is infinite.

J. Marshall Ash and Yiren Tan, A rational number of the form a^a with a irrational, Mathematical Gazette 96, March 2012, pp. 106-109.
Wikipedia, Tetration, Open questions
Wikipedia, Super-root

Cf. A225134 (decimal expansion), A225208 (Engel expansion), A153510 (second super-root of 2).

ContinuedFraction[FindRoot[x^x^x^x == 2, {x, 1}, WorkingPrecision -> 110][[1, 2]], 105]

Offset changed by Andrew Howroyd, Jul 07 2024

A225208 Engel expansion of the positive root of x^x^x^x = 2.

Original entry on oeis.org

1, 3, 3, 52, 106, 260, 279, 334, 491, 536, 728, 1161, 5678, 15183, 41437, 189034, 281965, 1118629, 3473978, 32869874, 82525851, 159312757, 424570638, 472381891, 563118608, 579529452, 1426303902, 2330077798, 2991863700, 25850322702, 34547004920, 37294688664

Offset: 1

Alois P. Heinz, May 01 2013

nonn

It is not known if the positive root of x^x^x^x = 2 is a rational number and, in consequence, whether this sequence is finite or not.

			1.44660143242986417459733398759766148...

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

Alois P. Heinz, Table of n, a(n) for n = 1..100
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner 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
Wikipedia, Tetration, Open questions
Index entries for sequences related to Engel expansions

Cf. A225134 (decimal expansion), A225153 (continued fraction).

Digits:= 500:
c:= solve(x^(x^(x^x))=2, x):
engel:= (r, n)-> `if`(n=0 or r=0, NULL, [ceil(1/r),
        engel(r*ceil(1/r)-1, n-1)][]):
engel(evalf(c), 39);

cp's OEIS Frontend

A225153 Continued fraction for the positive root of x^x^x^x = 2 (A225134).

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