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

%I A225208 #25 Feb 16 2025 08:33:19
%S A225208 1,3,3,52,106,260,279,334,491,536,728,1161,5678,15183,41437,189034,
%T A225208 281965,1118629,3473978,32869874,82525851,159312757,424570638,
%U A225208 472381891,563118608,579529452,1426303902,2330077798,2991863700,25850322702,34547004920,37294688664
%N A225208 Engel expansion of the positive root of x^x^x^x = 2.
%C A225208 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 A225208 F. Engel, Entwicklung der Zahlen nach Stammbrüchen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191.
%H A225208 Alois P. Heinz, <a href="/A225208/b225208.txt">Table of n, a(n) for n = 1..100</a>
%H A225208 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 A225208 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 A225208 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>
%H A225208 Wikipedia, <a href="https://en.wikipedia.org/wiki/Engel_expansion">Engel Expansion</a>
%H A225208 Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration#Open_questions">Tetration, Open questions</a>
%H A225208 <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>
%e A225208 1.44660143242986417459733398759766148...
%p A225208 Digits:= 500:
%p A225208 c:= solve(x^(x^(x^x))=2, x):
%p A225208 engel:= (r, n)-> `if`(n=0 or r=0, NULL, [ceil(1/r),
%p A225208         engel(r*ceil(1/r)-1, n-1)][]):
%p A225208 engel(evalf(c), 39);
%Y A225208 Cf. A225134 (decimal expansion), A225153 (continued fraction).
%K A225208 nonn
%O A225208 1,2
%A A225208 _Alois P. Heinz_, May 01 2013