A225208 -id:A225208 - cp's OEIS frontend

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

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

A225153 Continued fraction for the positive root of x^x^x^x = 2 (A225134).

Original entry on oeis.org

1, 2, 4, 5, 1, 1, 184, 1, 1, 8, 1, 7, 1, 12, 3, 1, 4, 2, 1, 2, 1, 125, 1, 2, 1, 1, 2, 2, 5, 12, 7, 1, 8, 2, 1, 6, 1, 3, 2, 1, 2, 1, 14, 1, 1, 1, 3, 1, 1, 6485, 1, 1, 1, 3, 1, 2, 1, 1, 1, 17, 1, 2, 3, 3, 3, 2, 7, 1, 2, 1, 8, 1, 9, 1, 1, 7, 1, 4, 9, 1, 1, 1, 1, 3, 2

Offset: 0

Vladimir Reshetnikov, Apr 30 2013

x = 1.44660143242986417... = 1 + 1/(2 + 1/(4 + 1/(5 + 1/(1 + 1/(1 + 1/(184 + 1/(...))))))).
This constant is sometimes called the 4th super-root of 2.
It is unknown if it is rational, algebraic irrational, or transcendental. Hence, it is unknown if this continued fraction is aperiodic, or even if it is infinite.

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
Wikipedia, Super-root

Cf. A225134 (decimal expansion), A225208 (Engel expansion), A153510 (second super-root of 2).

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

Offset changed by Andrew Howroyd, Jul 07 2024

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI