A030798 -id:A030798 - cp's OEIS frontend

A186503 Decimal expansion of the solution x to x^x = 13.

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 22 2011

			2.6410619164843958084118390040657912549308732...

Cf. A016636, A030798, A173158, A000038, A173159, A173160, A173161, A173162, A173163, A090625, A186501, A186502, A186504, A344930, A093590.

x=13;RealDigits[Log[x]/ProductLog[Log[x]],10,103][[1]]
RealDigits[x/.FindRoot[x^x==13,{x,2.6},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Feb 07 2025 *)

Equals log(13)/LambertW(log(13)). - Alois P. Heinz, Jun 16 2021

A186504 Decimal expansion of the solution x to x^x = 14.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Feb 22 2011

			2.6785234858912995813011990010099506157786992..

Cf. A030798, A173158, A000038, A173159, A173160, A173161, A173162, A173163, A090625, A186501, A186502, A186503, A344930, A093590.

x=14;RealDigits[Log[x]/ProductLog[Log[x]],10,200][[1]]
RealDigits[x/.FindRoot[x^x==14,{x,2},WorkingPrecision->120]][[1]] (* Harvey P. Dale, Aug 08 2023 *)

A344930 Decimal expansion of the real solution to x^x = 15.

Original entry on oeis.org

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

Offset: 1

Christoph B. Kassir, Jun 02 2021

2.7131636040042392095764012768285093718781...

Cf. A030798, A173158, A000038, A173159, A173160, A173161, A173162, A173163, A090625, A186501, A186502, A186503, A186504, A093590.

RealDigits[Log[15]/ProductLog[Log[15]], 10, 100][[1]] (* Amiram Eldar, Jun 02 2021 *)
RealDigits[x/.FindRoot[x^x==15,{x,2.7},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Apr 18 2022 *)

Equals log(15)/W(log(15)), where W(z) is the Lambert W Function.

A302973 Decimal expansion of the solution to (1 + x)*x^x = (1 - x)^x.

Original entry on oeis.org

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

Offset: 0

Juri-Stepan Gerasimov, Apr 16 2018

			0.2938600137873990375098351643534479...

Cf. A030798, A300304.

Take[RealDigits[x /. FindRoot[(1 + x)*x^x == (1 - x)^x, {x, 1/2}, WorkingPrecision -> 120], 10][[1]], 105] (* Vaclav Kotesovec, Apr 17 2018 *)

solve(x=1/10, 1, (1 + x)*x^x - (1 - x)^x) \\ Michel Marcus, Apr 16 2018

More digits from Alois P. Heinz, Apr 16 2018

A153510 Continued fraction for the positive solution of the equation x^x = 2.

Original entry on oeis.org

1, 1, 1, 3, 1, 2, 3, 1, 2, 1, 24, 2, 1, 52, 6, 4, 1, 3, 4, 2, 1, 14, 2, 5, 1, 7, 2, 2, 2, 1, 1, 1, 6, 3, 1, 1, 13, 1, 1, 1, 9, 1, 3, 1, 2, 9, 162, 1, 1, 195, 4, 1, 3, 47, 1, 1, 1, 1, 2, 1, 2, 1, 1, 5, 1, 1, 2, 4, 1, 1, 1, 2, 11, 1, 2, 7, 1, 2, 14, 1, 28, 1, 4, 1, 3, 1, 1, 1, 10, 2, 7, 5, 3, 1, 2, 4, 5, 35, 1

Offset: 0

Vladimir Reshetnikov, Dec 28 2008

Cf. A030798 (decimal expansion).

ContinuedFraction[Log[2]/ProductLog[Log[2]], 100]

Offset changed by Andrew Howroyd, Aug 03 2024

A173566 a(n+1) = a(n)^a(n), with a(1) = 2.

Original entry on oeis.org

2, 4, 256

Offset: 1

Franklin T. Adams-Watters, Aug 03 2011

The next term, a(4), is 2^2048, with 617 digits.
From Natan Arie Consigli, Dec 01 2015: (Start)
Possible other sequence with the same first three entries:
a(1) = 2;
a(2) = Triangle(2);
a(3) = Square(2);
a(4) = Pentagon(2);
etc., where, in Steinhaus-Moser notation,
Triangle(n) = n^n;
Square(n) = Triangle(Triangle...(n)...) (with n inside n nested triangles);
Pentagon(n) = Square(Square...(n)...)(with n inside n nested squares);
etc.
Start with a(1) = 2, a(2) = triangle(2) = 4, a(3) = square(2) = 256, a(4) = pentagon(4) = 256^^256 (power tower of 256s with height 256).
(End)

			a(3) = square(2) = triangle(triangle(2)) = triangle(2^2) = 4^4 = 256.
a(4) = 2^2048.
a(5) = 2^(2^2059).

Michel Marcus, Table of n, a(n) for n = 1..4
Wikipedia, Steinhaus-Moser notation.

Cf. A030798 ("preceding term"), A054874 (log base 2).

[n le 1 select 2 else Self(n-1)^Self(n-1): n in [1..4]]; // Vincenzo Librandi, Dec 17 2015

RecurrenceTable[{a[1] == 2, a[n] == a[n - 1]^a[n - 1]}, a, {n, 4}] (* Vincenzo Librandi, Dec 17 2015 *)

A199550 Decimal expansion of the positive root of x^x^x = 2.

Original entry on oeis.org

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

Offset: 1

Vladimir Reshetnikov, Nov 07 2011

As follows from Gelfond's theorem, the root is irrational, so this sequence is infinite and aperiodic. Its transcendence is, apparently, still an open problem. - Vladimir Reshetnikov, Apr 27 2013

			1.4766843373578699470892355853738898386551689309855269844644...

Ash J. Marshall and Yiren Tan, A rational number of the form a^a with a irrational, Mathematical Gazette 96, March 2012, pp. 106-109.
Eric Weisstein's World of Mathematics, Gelfond's Theorem

Cf. A030798.

First[RealDigits[Root[{Function[x, x^x^x - 2], 1.477`4}], 10, 100]]

solve(x=1,2,x^x^x-2) \\ Charles R Greathouse IV, Apr 14 2014

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

A194624 Decimal expansion of the smaller solution to x^x = 3/4.

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Sep 02 2011

Since (1/e)^(1/e) < 3/4 < 1, the equation x^x = 3/4 has two solutions x = a and x = b with 0 < a < 1/e < b < 1. Both solutions are transcendental (see Proposition 2.2 in Sondow-Marques 2010).

			0.15351678966395294715006833297846322771126948548996962031798542833437261364190...

J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164.
Index entries for transcendental numbers

Cf. A030798 (x^x = 2), A072364 ((1/e)^(1/e)), A194625 (larger solution to x^x = 3/4).

x = x /. FindRoot[x^x == 3/4, {x, 0.1}, WorkingPrecision -> 120]; RealDigits[x, 10, 100] // First

A194625 Decimal expansion of the larger solution to x^x = 3/4.

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Sep 02 2011

Since (1/e)^(1/e) < 3/4 < 1, the equation x^x = 3/4 has two solutions x = a and x = b with 0 < a < 1/e < b < 1. Both solutions are transcendental (see Proposition 2.2 in Sondow-Marques 2010).

			0.636262939294531019987513755204233173117867057936262294886540645406389214402799...

J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164.
Index entries for transcendental numbers

Cf. A030798 (x^x = 2), A072364 ((1/e)^(1/e)), A194624 (smaller solution to x^x = 3/4).

x = x /. FindRoot[x^x == 3/4, {x, 0.7}, WorkingPrecision -> 120]; RealDigits[x, 10, 100] // First

cp's OEIS Frontend

A186503 Decimal expansion of the solution x to x^x = 13.

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A186504 Decimal expansion of the solution x to x^x = 14.

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A344930 Decimal expansion of the real solution to x^x = 15.

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A302973 Decimal expansion of the solution to (1 + x)*x^x = (1 - x)^x.

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A153510 Continued fraction for the positive solution of the equation x^x = 2.

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A173566 a(n+1) = a(n)^a(n), with a(1) = 2.

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A199550 Decimal expansion of the positive root of x^x^x = 2.

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

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A194624 Decimal expansion of the smaller solution to x^x = 3/4.

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