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Engel expansion of the positive root of x^x^x^x = 2.
3

%I #24 Apr 13 2018 11:31:04

%S 1,3,3,52,106,260,279,334,491,536,728,1161,5678,15183,41437,189034,

%T 281965,1118629,3473978,32869874,82525851,159312757,424570638,

%U 472381891,563118608,579529452,1426303902,2330077798,2991863700,25850322702,34547004920,37294688664

%N Engel expansion of the positive root of x^x^x^x = 2.

%C 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.

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

%H Alois P. Heinz, <a href="/A225208/b225208.txt">Table of n, a(n) for n = 1..100</a>

%H F. Engel, <a href="/A006784/a006784.pdf">Entwicklung der Zahlen nach Stammbruechen</a>, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.

%H P. Erdős and Jeffrey Shallit, <a href="http://www.numdam.org/item?id=JTNB_1991__3_1_43_0">New bounds on the length of finite Pierce and Engel series</a>, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration#Open_questions">Tetration, Open questions</a>

%H <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>

%e 1.44660143242986417459733398759766148...

%p Digits:= 500:

%p c:= solve(x^(x^(x^x))=2, x):

%p engel:= (r, n)-> `if`(n=0 or r=0, NULL, [ceil(1/r),

%p engel(r*ceil(1/r)-1, n-1)][]):

%p engel(evalf(c), 39);

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

%K nonn

%O 1,2

%A _Alois P. Heinz_, May 01 2013