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

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

Author

Alois P. Heinz, May 01 2013

Keywords

• nonn

Comments

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.

Examples

			1.44660143242986417459733398759766148...
		

References

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

Links

• 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

Crossrefs

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

Programs

Maple

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);