A073226 -id:A073226 - cp's OEIS frontend

A064107 Continued fraction quotients for e^e = 15.15426224... (A073226).

15, 6, 2, 13, 1, 3, 6, 2, 1, 1, 5, 1, 1, 1, 9, 4, 1, 1, 1, 6, 7, 1, 2, 4, 1, 2, 2, 24, 1, 2, 4, 56, 1, 1, 2, 4, 1, 75, 1, 5, 1, 2, 2, 1, 137, 2, 2, 97, 3, 16, 1, 1, 1, 1, 3, 5, 12, 1, 1, 2, 1, 53, 1, 2, 5, 3, 2, 4, 1, 2, 1, 39, 1, 2, 1, 4, 1, 11, 1, 5, 5, 1, 4, 1, 17, 12, 4, 82, 1, 4, 6, 25, 3, 2, 3

Offset: 0

Labos Elemer, Sep 17 2001

It was conjectured (but remains unproved) that this sequence is infinite and aperiodic, but it is difficult to determine who first posed this problem. - Vladimir Reshetnikov, Apr 27 2013

			15.154262241479264189760430... = 15 + 1/(6 + 1/(2 + 1/(13 + 1/(1 + ...)))). - _Harry J. Smith_, Apr 30 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
Eric Weisstein's World of Mathematics, Transcendental Number.
Wikipedia, List of unsolved problems in mathematics, Analysis.
Wikipedia, Irrational number, Open questions.

Cf. A058287, A058288, A073226 (decimal expansion), A159825.

with(numtheory): cfrac(evalf((exp(1))^(exp(1)),2560),256,'quotients');

ContinuedFraction[E^E,100] (* Harvey P. Dale, Sep 29 2012 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(exp(exp(1))); for (n=1, 20000, write("b064107.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 30 2009

Offset changed by Andrew Howroyd, Aug 05 2024

A073229 Decimal expansion of e^(1/e).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jul 22 2002

e^(1/e) = 1/((1/e)^(1/e)) (reciprocal of A072364).
Let w(n+1)=A^w(n); then w(n) converges if and only if (1/e)^e <= A <= e^(1/e) (see the comments in A073230) for initial value w(1)=A. If A=e^(1/e) then lim_{n->infinity} w(n) = e. - Benoit Cloitre, Aug 06 2002; corrected by Robert FERREOL, Jun 12 2015
x^(1/x) is maximum for x = e and the maximum value is e^(1/e). This gives an interesting and direct proof that 2 < e < 4 as 2^(1/2) < e^(1/e) > 4^(1/4) while 2^(1/2) = 4^(1/4). - Amarnath Murthy, Nov 26 2002
For large n, A234604(n)/A234604(n-1) converges to e^(1/e). - Richard R. Forberg, Dec 28 2013
Value of the unique base b > 0 for which the exponential curve y=b^x and its inverse y=log_b(x) kiss each other; the kissing point is (e,e). - Stanislav Sykora, May 25 2015
Actually, there is another base with such property, b=(1/e)^e with kiss point (1/e,1/e). - Yuval Paz, Dec 29 2018
The problem of finding the maximum of f(x) = x^(1/x) was posed and solved by the Swiss mathematician Jakob Steiner (1796-1863) in 1850. - Amiram Eldar, Jun 17 2021

			1.44466786100976613365833910859...

David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 35.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Ovidiu Furdui, Problem 11982, The American Mathematical Monthly, Vol. 124, No. 5 (2017), p. 465; A Limit of a Power of a Sum, Solution to Problem 11982 by Roberto Tauraso, ibid., Vol. 126, No. 2 (2019), p. 187.
Simon Plouffe, exp(1/e).
Jonathan Sondow and Diego Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae, Vol. 37 (2010), pp. 151-164; see Definition 4.1 on p. 158.
Jacob Steiner, Über das größte Product der Theile oder Summanden jeder Zahl, Crelle, Vol. 40 (1850), pp. 208; alternative link.
Eric Weisstein's World of Mathematics, Steiner's Problem.

Cf. A001113 (e), A068985 (1/e), A073230 ((1/e)^e), A072364 ((1/e)^(1/e)), A073226 (e^e).
Cf. A093157, A103476.
Cf. A038051, A086331, A252782, A270593, A270917, A270923, A277473, A309652, A332408.

evalf[110](exp(exp(-1))); # Muniru A Asiru, Dec 29 2018

Mathematica
```
RealDigits[ E^(1/E), 10, 110] [[1]]
```
PARI
```
exp(1)^exp(-1)
```

Equals 1 + Integral_{x = 1/e..1} (1 + log(x))/x^x dx = 1 - Integral_{x = 0..1/e} (1 + log(x))/x^x dx. - Peter Bala, Oct 30 2019
Equals Sum_{k>=0} exp(-k)/k!. - Amiram Eldar, Aug 13 2020
Equals lim_{x->oo} (Sum_{n>=1} (x/n)^n)^(1/x) (Furdui, 2017). - Amiram Eldar, Mar 26 2022

A073233 Decimal expansion of Pi^Pi.

Original entry on oeis.org

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

Offset: 2

Rick L. Shepherd, Jul 21 2002

A weak form of Schanuel's Conjecture implies that Pi^Pi is transcendental--see Marques and Sondow (2012).

			36.4621596072079117709908260226...

Harry J. Smith, Table of n, a(n) for n = 2..20000
D. Marques and J. Sondow, The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture, arXiv:1212.6931 [math.NT], 2012-2013.

Cf. A000796 (Pi), A073234 (Pi^Pi^Pi), A073237 (ceil(Pi^Pi^...^Pi), n Pi's), A073238 (Pi^(1/Pi)), A073239 ((1/Pi)^Pi), A073240 ((1/Pi)^(1/Pi)), A073243 (limit of (1/Pi)^(1/Pi)^...^(1/Pi)), A073236 (Pi analog of A004002).
Cf. A073226 (e^e).
Cf. A049006 (i^i), A116186 (real part of i^i^i).
Cf. A194555 (real part of i^e^Pi).

RealDigits[N[Pi^Pi,200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

PARI
```
Pi^Pi
```

{ default(realprecision, 20080); x=Pi^Pi/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b073233.txt", n, " ", d)); } \\ Harry J. Smith, Apr 30 2009

A073230 Decimal expansion of (1/e)^e.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 22 2002

(1/e)^e = e^(-e) = 1/(e^e) (reciprocal of A073226).

The power tower function f(x)=x^(x^(x^...)) is defined on the closed interval [e^(-e),e^(1/e)]. - Lekraj Beedassy, Mar 17 2005

			0.06598803584531253707679018759...

Paul Halmos, "Problems for Mathematicians, Young and Old", Dolciani Mathematical Expositions, 1991, Solution to problem 8A (Power Tower) p. 240.

G. C. Greubel, Table of n, a(n) for n = 0..10000
J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; see the Appendix.

Cf. A001113 (e), A068985 (1/e), A073229 (e^(1/e)), A072364 ((1/e)^(1/e)), A073226 (e^e).

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

a=IntegerDigits[IntegerPart[(1/E)^E*10^99]];PrependTo[a,0] (* Vladimir Joseph Stephan Orlovsky, Jul 17 2008 *)
RealDigits[(1/E)^E, 10, 100][[1]] (* Alonso del Arte, Aug 26 2011 *)

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

A085667 Decimal expansion of e^e^e^e.

Original entry on oeis.org

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

Offset: 1656521

N. J. A. Sloane, Jul 15 2003

			2.331504399007195462289689911012137666332017428963... * 10^1656520

J. Blanck, Exact real arithmetic systems: results of competition, pp. 389-393 of J. Blanck et al., eds., Computability and Complexity in Analysis (CCA 2000), Lect. Notes Computer Science, Springer-Verlag, 2001.
Hans Havermann, integer part of e^e^e^e (in blocks of 100 digits) = the first 1656521 terms
Hans Havermann, fractional part of e^e^e^e (in blocks of 100 digits) = an additional 1656521 terms

Cf. A073226, A073227, A073228, A181180, A073232, A202955.

RealDigits[Exp[Exp[Exp[Exp[1]]]],10,120][[1]] (* Harvey P. Dale, Sep 06 2012 *)

exp(exp(exp(exp(1)))) \\ Charles R Greathouse IV, Mar 25 2014

A073227 Decimal expansion of e^e^e.

Original entry on oeis.org

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

Offset: 7

Rick L. Shepherd, Jul 22 2002

A weak form of Schanuel's Conjecture implies that e^e^e is transcendental--see Marques and Sondow (2012).

			3814279.10476022059220921959409...

Harry J. Smith, Table of n, a(n) for n = 7..20000
D. Marques and J. Sondow, The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture, arXiv:1212.6931 [math.NT], 2013.

Cf. A001113 (e), A073226 (e^e), A004002 (e^e^...^e, n times, rounded), A073228 ((e^e)^e), A073231 ((1/e)^(1/e)^(1/e)).

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

RealDigits[E^E^E,10,120][[1]] (* Harvey P. Dale, Dec 14 2011 *)

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

{ default(realprecision, 20080); x=exp(exp(exp(1)))/1000000; for (n=7, 20000, d=floor(x); x=(x-d)*10; write("b073227.txt", n, " ", d)); } \\ Harry J. Smith, Apr 30 2009

A073244 Decimal expansion of Pi - e.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 21 2002

			0.42331082513074800310235591192...

Cf. A059742 (Pi+e), A000796 (Pi), A001113 (e), A019609 (Pi*e), A061382 (Pi/e), A061360 (e/Pi), A039661 (e^Pi), A059850 (Pi^e), A073233 (Pi^Pi), A073226 (e^e), A049006 (i^i = e^(-Pi/2)).

Cf. A110564 for continued fraction for Pi - e.

RealDigits[N[Pi-E,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

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

A194556 Decimal expansion of (9/4)^(27/8) = (27/8)^(9/4).

Original entry on oeis.org

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

Offset: 2

Jonathan Sondow, Aug 30 2011

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 = 2 gives (x,y) = (9/4,27/8). See Sondow and Marques 2010, pp. 155-157.

(9/4)^(27/8) = (27/8)^(9/4) corresponds to (4/9)^(4/9) = (8/27)^(8/27) (see A194789) under the equivalence x^y = y^x <==> (1/x)^(1/x) = (1/y)^(1/y).

			15.438887358552583183604460013074909718871494279680272412854330453294418363...

J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae, 37 (2010), 151-164.
Index entries for algebraic numbers, degree 4

Cf. A073226 (e^e), A194557 (sqrt(3)^sqrt(27) = sqrt(27)^sqrt(3)), A194789 ((4/9)^(4/9) = (8/27)^(8/27)).

RealDigits[ (9/4)^(27/8), 10, 100] // First

-((9*ProductLog(-1, -(4/9)*log(9/4)))/(4*log(9/4))), where ProductLog is the Lambert W function, simplifies to 27/8. - Jean-François Alcover, Jun 01 2015

A056072 a(n) = floor(e^e^ ... ^e), with n e's.

Original entry on oeis.org

1, 2, 15, 3814279

Offset: 0

Robert G. Wilson v, Jul 26 2000

nonn

The next term is too large to include.
From Vladimir Reshetnikov, Apr 27 2013: (Start)
a(4) = 2331504399007195462289689911...2579139884667434294745087021 (1656521 decimal digits in total), given by initial segment of A085667.
a(5) has more than 10^10^6 decimal digits.
a(6) has more than 10^10^10^6 decimal digits. (End)

Cf. A001113, A004002, A003417, A064107, A159825, A073226, A073227, A085667, A096232, A225064, A225053.

p:= n-> `if`(n=0, 1, exp(1)^p(n-1)):
a:= n-> floor(p(n)):
seq(a(n), n=0..3);  # Alois P. Heinz, Jul 20 2024

Floor[NestList[Exp, 1, 3]] (* Vladimir Reshetnikov, Apr 29 2013 *)

A056072(n,f=floor)=f(exp(if(n>0,A056072(n-1,x->x))))  \\ M. F. Hasler, May 01 2013

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

Jonathan Sondow, Aug 30 2011

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.

			17.361905250953135215415714826833267582295532184890864078454696057446763745...

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

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

-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

cp's OEIS Frontend

A064107 Continued fraction quotients for e^e = 15.15426224... (A073226).

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A073229 Decimal expansion of e^(1/e).

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A073233 Decimal expansion of Pi^Pi.

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A073230 Decimal expansion of (1/e)^e.

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A085667 Decimal expansion of e^e^e^e.

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A073227 Decimal expansion of e^e^e.

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A073244 Decimal expansion of Pi - e.

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