cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-25 of 25 results.

A333318 Decimal expansion of the number x that satisfies x^x = 2x as well as x != 2.

Original entry on oeis.org

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

Views

Author

Wyatt Porter, Mar 14 2020

Keywords

Examples

			0.34632336227858092206485655218088677211354545468282...

Crossrefs

Cf. A030798.

Programs

  • Mathematica
    RealDigits[x/.FindRoot[-2 x + x^x == 0, {x, 1/2}, WorkingPrecision -> 120]][[1]] (* Michael De Vlieger, Mar 14 2020 *)
  • PARI
    solve(x=0.1, 1, x^x-2*x) \\ Michel Marcus, Mar 15 2020

Formula

x^x = 2x, x != 2.

A344905 Decimal expansion of the solution to x^x = sqrt(2).

Original entry on oeis.org

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

Views

Author

Christoph B. Kassir, Jun 01 2021

Keywords

Examples

			1.304351178901036533647201231486234...

Crossrefs

Cf. A030798, A073084.

Programs

  • Mathematica
    RealDigits[Log[Sqrt[2]]/ProductLog[Log[Sqrt[2]]], 10, 100][[1]] (* Amiram Eldar, Jun 02 2021 *)
RealDigits[x/.FindRoot[x^x==Sqrt[2],{x,1},WorkingPrecision-> 120],10,120][[1]] (* Harvey P. Dale, Jun 18 2021 *)
  • PARI
    solve(x=1,2,x^x-sqrt(2)) \\ Hugo Pfoertner, Jun 02 2021

Formula

Equals log(2)/(2*LambertW(log(2)/2)). - Alois P. Heinz, Jun 02 2021
Equals 1/A073084. - Jason Bard, Aug 20 2025

A378960 The "tetrational mean" of 2 and 3 determined as the mutual limit of interdependent sequences.

Original entry on oeis.org

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

Views

Author

Pham G. Hoang, Dec 12 2024

Keywords

Comments

In an attempt to generalize the arithmetic mean (sum-based) and the geometric mean (product-based) to a similar construct for exponentiation, one can devise a simple definition using 2 interdependent sequences:
a_0 = x, b_0 = y,
a_n = exp(LambertW(log(a_{n-1}^b_{n-1}))),
b_n = exp(LambertW(log(b_{n-1}^a_{n-1}))), where x and y are the numbers for which we have to determine their "tetrational mean".
The averaging operation is the square super-root of each of the 2 possible exponentiation orders to give out the successive term of each defining sequence. The square super-root of x is exp(LambertW(log(x))) for a particular branch of the LambertW function.
If a_n and b_n converge to a number C then the "tetrational mean" of x and y is C. There may be a need to choose a particular branch of the LambertW function depending on the values of x and y (and that of log(x^y) and log(y^x)). This constant is based on the principal branch of the LambertW function.

Examples

			2.419365321844921788860745468932754354...

Links

Crossrefs

Cf. A068985, A030798, A010464.

A173169 Decimal expansion of the solution x to x^x = A, the Glaisher-Kinkelin constant (A074962).

Original entry on oeis.org

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

Views

Author

Vladimir Joseph Stephan Orlovsky, Feb 11 2010

Keywords

Examples

			1.22512630432118191..^1.22512630432118191.. = 1.28242712910062263687534256886979..

Crossrefs

Cf. A030437, A030797, A030798, A090625, A133930, A173158-A173166.

Programs

  • Mathematica
    x=Glaisher;RealDigits[Log[x]/ProductLog[Log[x]],10,4*5! ][[1]]
  • PARI
    (x->x/lambertw(x))(1/12-zeta'(-1)) \\ Charles R Greathouse IV, Dec 12 2013

Extensions

Keyword:cons added by R. J. Mathar, Feb 13 2010

A305187 Decimal expansion of the solution to x^x^x = 3.

Original entry on oeis.org

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

Views

Author

Juri-Stepan Gerasimov, May 27 2018

Keywords

Comments

Let x(m) be the solution to the equation x^x^x^...^x = m, where x appears m times on the left hand side; e.g.,
decimal
m equation solution x(m) expansion
==== ==================== ============= =============
1 x = 1 1.00000000... A000007
2 x^x = 2 1.55961046... A030798
3 x^x^x = 3 1.63507847... this sequence
4 x^x^x^x = 4 1.62036995...
5 x^x^x^x^x = 5 1.59340881...
6 x^x^x^x^x^x = 6 1.56864406...
7 x^x^x^x^x^x^x = 7 1.54828598...
.
10 x^x^x^x^...^x = 10 1.50849792...
.
100 x^x^x^x^...^x = 100 1.44567285...
.
1000 x^x^x^x^...^x = 1000 1.44467831...
.
Then x(1) < x(m) < x(3) for all m >= 4.
Let y(k/2) be the solution to the equation y^y^y^...^y = (k/2)*y^y, where y appears k times on the left hand side; e.g.,
decimal
k equation solution y(k/2) expansion
= ========================= =============== =========
1 y = (1/2)*y^y 2 A000038
2 y^y = (2/2)*y^y indeterminate
3 y^y^y = (3/2)*y^y 1.6998419085...
4 y^y^y^y = (4/2)*y^y 1.6396207046...
5 y^y^y^y^y = (5/2)*y^y 1.5987769216...
6 y^y^y^y^y^y = (6/2)*y^y 1.5694666408...
7 y^y^y^y^y^y^y = (7/2)*y^y 1.5476452822...
.
What is lim_{k -> infinity} y(k/2)?
Lim_{m -> infinity} x(m) = e^(1/e). - Jon E. Schoenfield, Jul 23 2018
Lim_{k -> infinity} y(k/2) = e^(1/e). - Jon E. Schoenfield, Aug 01 2018

Examples

			1.635078474636375245899757198787500888...

Crossrefs

Cf. A000007, A000038, A030798.

Programs

  • Mathematica
    RealDigits[ FindRoot[ x^x^x == 3, {x, 1}, WorkingPrecision -> 128][[1, 2]]][[1]] (* Robert G. Wilson v, Jun 13 2018 *)
  • PARI
    default(realprecision,333);
solve(x=1.6, 1.7, x^x^x-3) \\ Joerg Arndt, May 27 2018

Extensions

More digits from Michel Marcus, Joerg Arndt, May 27 2018
Previous Showing 21-25 of 25 results.

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