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.

Showing 1-10 of 10 results.

A194557 Decimal expansion of sqrt(3)^sqrt(27) = sqrt(27)^sqrt(3).

Original entry on oeis.org

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

Views

Author

Jonathan Sondow, Aug 30 2011

Keywords

Comments

Positive real numbers x < y with x^y = y^x are parameterized by (x,y) = ((1 + 1/t)^t,(1 + 1/t)^(t+1)) for t > 0. For example, t = 1/2 gives (x,y) = (sqrt(3),sqrt(27)). See Sondow and Marques 2010, pp. 155-157.

Examples

			17.361905250953135215415714826833267582295532184890864078454696057446763745...

Links

Crossrefs

Cf. A073226 (decimal expansion of e^e), A194556 (decimal expansion of (9/4)^(27/8) = (27/8)^(9/4)).

Programs

  • Mathematica
    RealDigits[ Sqrt[3]^Sqrt[27], 10, 100] // First

Formula

-2*sqrt(3)*ProductLog(-1, -log(3)/(2*sqrt(3)))/log(3), where ProductLog is the Lambert W function, simplifies to sqrt(27). - Jean-François Alcover, Jun 01 2015

A194789 Decimal expansion of (4/9)^(4/9) = (8/27)^(8/27).

Original entry on oeis.org

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

Views

Author

Jonathan Sondow, Sep 02 2011

Keywords

Comments

(4/9)^(4/9) = (8/27)^(8/27) corresponds to (9/4)^(27/8) = (27/8)^(9/4) (see A194556) under the equivalence x^x = y^y <==> (1/x)^(1/y) = (1/y)^(1/x).

Examples

			0.6973879457621430704075323673549042345770355186105029836376253048204852032246954...

Links

Crossrefs

Cf. A194556.

Programs

  • Mathematica
    RealDigits[(4/9)^(4/9), 10, 100] // First

A194622 Decimal expansion of x with 0 < x < y and x^y = y^x = 17.

Original entry on oeis.org

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

Views

Author

Jonathan Sondow, Aug 30 2011

Keywords

Comments

Given z > 0, there exist positive real numbers x < y with x^y = y^x = z, if and only if z > e^e. In that case, (x,y) = ((1 + 1/t)^t,(1 + 1/t)^(t+1)) for some t > 0. For example, t = 1 gives 2^4 = 4^2 = 16 > e^e. When x^y = y^x = 17, at least one of x and y is transcendental. See Sondow and Marques 2010, pp. 155-157.

Examples

			x=1.7838142517704619219012767113132837917073658346795118208782477687564285462224371...

Links

Crossrefs

Cf. A073226 (e^e), A194556 ((9/4)^(27/8) = (27/8)^(9/4)), A194557 (sqrt(3)^sqrt(27) = sqrt(27)^sqrt(3)), A194623 (y with 0 < x < y and x^y = y^x = 17).

Programs

  • Mathematica
    x[t_] := (1 + 1/t)^t; y[t_] := (1 + 1/t)^(t + 1); t = t/. FindRoot[ x[t]^y[t] == 17, {t, 1}, WorkingPrecision -> 120]; RealDigits[ x[t], 10, 100] // First

A194623 Decimal expansion of y with 0 < x < y and x^y = y^x = 17.

Original entry on oeis.org

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

Views

Author

Jonathan Sondow, Aug 30 2011

Keywords

Comments

Given z > 0, there exist positive real numbers x < y with x^y = y^x = z, if and only if z > e^e. In that case, (x,y) = ((1 + 1/t)^t,(1 + 1/t)^(t+1)) for some t > 0. For example, t = 1 gives 2^4 = 4^2 = 16 > e^e. When x^y = y^x = 17, at least one of x and y is transcendental. See Sondow and Marques 2010, pp. 155-157.

Examples

			y=4.89536795554611347196719338722983584947273195280937244363084664929554121...

Links

Crossrefs

Cf. A073226 (e^e), A194556 ((9/4)^(27/8) = (27/8)^(9/4)), A194557 (sqrt(3)^sqrt(27) = sqrt(27)^sqrt(3)), A194622 (x with 0 < x < y and x^y = y^x = 17).

Programs

  • Mathematica
    x[t_] := (1 + 1/t)^t; y[t_] := (1 + 1/t)^(t + 1); t = t/. FindRoot[x[t]^y[t] == 17, {t, 1}, WorkingPrecision -> 120]; RealDigits[y[t], 10, 100] // First

A258500 Decimal expansion of the nontrivial real solution of x^(3/2) = (3/2)^x.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 01 2015

Keywords

Examples

			x0 = 7.408764686965774521957295028510614389804171141074...
z = x0^(3/2) = 20.16595073003535058942970947434890012034363496 ...
z > e^e = 15.15426224... = A073226.

Links

Crossrefs

Cf. A073226, A194556, A194557, A258501 (x^(5/2)=(5/2)^x), A258502 (x^(7/2)=(7/2)^x).

Programs

  • Mathematica
    x0 = -((x*ProductLog[-1, -(Log[x]/x)])/Log[x]) /. x -> 3/2; RealDigits[x0, 10, 101] // First
RealDigits[x/.FindRoot[x^(3/2)==(3/2)^x,{x,7},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Dec 07 2024 *)

Formula

x0 = -((x*ProductLog(-1, -(log(x)/x)))/log(x)), with x = 3/2, where ProductLog is the Lambert W function.

A258501 Decimal expansion of the nontrivial real solution of x^(5/2) = (5/2)^x.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 01 2015

Keywords

Examples

			x0 = 2.97028705025575877937998429103168637323950439632715025453459...
z = x0^(5/2) = 15.20533715980107653442006557792026842686895921352582...
z > e^e = 15.15426224... = A073226.

Links

Crossrefs

Cf. A073226, A194556, A194557, A258500 (x^(3/2)=(3/2)^x), A258502 (x^(7/2)=(7/2)^x).

Programs

  • Mathematica
    x0 = -((x*ProductLog[-1, -(Log[x]/x)])/Log[x]) /. x -> 5/2; RealDigits[x0, 10, 102] // First

Formula

x0 = -((x*ProductLog(-1, -(log(x)/x)))/log(x)), with x = 5/2, where ProductLog is the Lambert W function.

A258502 Decimal expansion of the nontrivial real solution of x^(7/2) = (7/2)^x.

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 01 2015

Keywords

Examples

			x0 = 2.189697551175613504808316814457313054952031983651039793...
z = x0^(7/2) = 15.53618787439250843837688346448101455506861788472622...
z > e^e = 15.15426224... = A073226.

Links

Crossrefs

Cf. A073226, A194556, A194557, A258500 (x^(3/2)=(3/2)^x), A258501 (x^(5/2)=(5/2)^x).

Programs

  • Mathematica
    x0 = -((x*ProductLog[-(Log[x]/x)])/Log[x]) /. x -> 7/2; RealDigits[x0, 10, 101] // First
RealDigits[x/.FindRoot[x^(7/2)==(7/2)^x,{x,2},WorkingPrecision-> 120]][[1]] (* Harvey P. Dale, Apr 19 2019 *)

Formula

x0 = -((x*ProductLog(-(log(x)/x)))/log(x)), with x = 7/2, where ProductLog is the Lambert W function.

A258503 Decimal expansion of (64/27)^(256/81) = (256/81)^(64/27).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 01 2015

Keywords

Examples

			15.2969313436178682300313080466454951313357722002517312514576871...

Links

Crossrefs

Cf. A194556, A194789, A258504.

Programs

  • Mathematica
    RealDigits[(64/27)^(256/81), 10, 103] // First

Formula

-((x*ProductLog(-1, -(log(x)/x)))/log(x)), replacing x with 64/27, gives 256/81 (ProductLog is the Lambert W function).

A258504 Decimal expansion of (27/64)^(27/64) = (81/256)^(81/256).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Jun 01 2015

Keywords

Examples

			0.6948233607279167955009391708983145471565914206815539940260405544...

Links

Crossrefs

Cf. A194556, A194789, A258503.

Programs

  • Mathematica
    RealDigits[(27/64)^(27/64), 10, 105] // First

Formula

x^x = y^y <==> (1/x)^(1/y) = (1/y)^(1/x), hence, from A258503:
(64/27)^(256/81) = (256/81)^(64/27) <==> (27/64)^(27/64) = (81/256)^(81/256).

A329458 Decimal expansion of x such that x^x * log(x) - x^x + 1 = 0, x > 1.

Original entry on oeis.org

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

Views

Author

Rick Novile, Nov 16 2019

Keywords

Comments

Equivalent to the coordinates of the self-intersection point of the graph y^x - x^y = y - x, where x, y > 1.

Examples

			x = 2.41173993056055928114518919802446413261177356034046370153515467138607...

Crossrefs

Cf. A073226 (e^e), A194622, A194623, A194556, A194557.

Programs

  • Mathematica
    FindRoot[x^x Log[x] - x^x + 1 == 0, {x, 2.40737, 2.41474}, WorkingPrecision -> 1000]
  • PARI
    solve(x=2, 3, x^x * log(x) - x^x + 1) \\ Michel Marcus, Nov 16 2019
  • PARI
    solve(x=2, 3, x - exp(1-1/x^x)) \\ Michel Marcus, Jul 14 2020

Formula

x^x * log(x) - x^x + 1 = 0; x != 1.
y^x - x^y = y - x; y = x; x != 1.
x = exp(1-x^(-x)); x > 1. - Rick Novile, Jul 14 2020
Showing 1-10 of 10 results.

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