A073226 -id:A073226 - cp's OEIS frontend

A073228 Decimal expansion of (e^e)^e.

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

Offset: 4

Rick L. Shepherd, Jul 22 2002

			1618.17799191265350166869122548...

G. C. Greubel, Table of n, a(n) for n = 4..10000

Cf. A001113 (e), A073226 (e^e), A073227 (e^e^e), A073231 ((1/e)^(1/e)^(1/e)), A073232 (((1/e)^(1/e))^(1/e)).

Exp(Exp(1))^Exp(1); // G. C. Greubel, May 29 2018

RealDigits[(E^E)^E, 10, 120][[1]] (* Alonso del Arte, Jul 03 2012 *)

PARI
```
exp(exp(1))^exp(1)
```

(e^e)^e = e^e^2. - Franklin T. Adams-Watters, Jun 20 2014

A234473 Decimal expansion of exp(exp(1)-1).

Original entry on oeis.org

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

Offset: 1

Richard R. Forberg, Dec 26 2013

Derived from an infinite sum of the Bell numbers (see formula below).

May also be written as exp(exp(1))/exp(1) = e^e/e.

			5.5749415247608806239669759227404843057060930975947002119298237838570...

Michael Ian Shamos, Shamos's catalog of the real numbers, (2011).

Cf. A000110, A001113 (e), A073226 (e^e), A274169.

RealDigits[Exp[E-1], 10, 100][[1]] (* Amiram Eldar, Dec 04 2021 *)

PARI
```
exp(exp(1)-1)
```

Equals Sum_{n>=0} Bell(n)/n!, where Bell(n) = A000110(n).

Equals Product_{k=1..oo} exp(1/k!). - Christoph B. Kassir, Dec 04 2021

A016066 a(n) = round(e^(e^n)).

Original entry on oeis.org

3, 15, 1618, 528491311, 514843556263457213182266, 28511235679461510605581038657982805983853648817939444953417128837

Offset: 0

Robert G. Wilson v

a(6) is a 176-digit number. - Jon E. Schoenfield, Sep 04 2017

			a(0) = round(e) = round(2.718281828459...) = 3.
a(1) = round(e^e) = round(15.154262241479264...) = 15.
a(2) = round(e^(e^2)) = round(1618.1779919126535...) = 1618.

Cf. A001113, A073226, A096232, A096404.

Table[Round[E^(E^n)], {n, 0, 5}] (* Harvey P. Dale, Oct 21 2016 *)

Corrected by N. J. A. Sloane, Aug 16 2004

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

Jonathan Sondow, Aug 30 2011

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.

			x=1.7838142517704619219012767113132837917073658346795118208782477687564285462224371...

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

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).

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

Jonathan Sondow, Aug 30 2011

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.

			y=4.89536795554611347196719338722983584947273195280937244363084664929554121...

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

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).

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

Jean-François Alcover, Jun 01 2015

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

Jonathan Sondow, Diego Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164.

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

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 *)

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

Jean-François Alcover, Jun 01 2015

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

Jonathan Sondow, Diego Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164.

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

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

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

Jean-François Alcover, Jun 01 2015

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

Jonathan Sondow, Diego Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164.

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

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 *)

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

A334399 Decimal expansion of sinh(e).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 26 2020

			(e^e - e^(-e))/2 = 7.54413710281697582634182004251653274...

Cf. A001113, A073226, A073230, A073742, A085659, A334400.

Mathematica
```
RealDigits[Sinh[E], 10, 110] [[1]]
```

Equals Sum_{k>=0} e^(2*k+1)/(2*k+1)!.

A334400 Decimal expansion of cosh(e).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 26 2020

			(e^e + e^(-e))/2 = 7.6101251386622883634186102301133791...

Cf. A001113, A073226, A073230, A073743, A085660, A334399.

Mathematica
```
RealDigits[Cosh[E], 10, 110] [[1]]
```

Equals Sum_{k>=0} e^(2*k)/(2*k)!.

cp's OEIS Frontend

A073228 Decimal expansion of (e^e)^e.

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A234473 Decimal expansion of exp(exp(1)-1).

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A016066 a(n) = round(e^(e^n)).

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

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

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

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

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

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