A225153 -id:A225153 - cp's OEIS frontend

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

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);

A225134 Decimal expansion of the positive root of x^x^x^x = 2.

Original entry on oeis.org

1, 4, 4, 6, 6, 0, 1, 4, 3, 2, 4, 2, 9, 8, 6, 4, 1, 7, 4, 5, 9, 7, 3, 3, 3, 9, 8, 7, 5, 9, 7, 6, 6, 1, 4, 8, 0, 6, 8, 7, 3, 2, 1, 0, 4, 2, 2, 8, 2, 2, 8, 0, 0, 2, 6, 3, 6, 3, 9, 0, 4, 7, 7, 2, 0, 9, 8, 5, 7, 0, 7, 6, 5, 9, 8, 3, 1, 0, 1, 6, 1, 4, 7, 4, 9, 2, 3, 5, 7, 2, 0, 0, 8, 1, 0, 9, 7, 6, 3, 0, 9, 9, 7, 5, 3

Offset: 1

Vladimir Reshetnikov, Apr 29 2013

It is unknown if this root is rational, algebraic irrational, or transcendental.

			1.4466014324298641745973339875976614806873210422822800263639...

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

Cf. A030798, A199550, A225153 (continued fraction), A225208 (Engel expansion).

RealDigits[FindRoot[x^x^x^x == 2, {x, 1}, WorkingPrecision -> 110][[1,2]], 10, 105][[1]]

solve(x=1,2,x^x^x^x-2) \\ Charles R Greathouse IV, Apr 15 2014

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple