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

1,2

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.

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

Example

1.44660143242986417459733398759766148...

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);

Crossrefs

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

Sequence in context: A265717 A292753 A268136 * A290567 A339758 A100065

Adjacent sequences: A225205 A225206 A225207 * A225209 A225210 A225211

Keyword

nonn

Author

Alois P. Heinz, May 01 2013

Status

approved