A073227 -id:A073227 - cp's OEIS frontend

A159825 Continued fraction for e^e^e A073227.

3814279, 9, 1, 1, 4, 1, 53, 26, 1, 13, 3, 1, 1, 22, 1, 226, 1, 5, 2, 1, 6, 2, 3, 1, 4, 1, 6, 39, 2, 1, 3, 1, 5, 1, 4, 1, 3, 1, 4, 1, 1, 19, 1, 2, 8899, 5, 2, 2, 1, 3, 3, 2, 2, 2, 1, 1, 3, 5, 1, 6, 10, 2, 1, 2, 1, 1, 1, 2, 2, 4, 1, 10, 2, 6, 1, 5, 6, 2, 4, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 11, 7, 3, 1, 4, 4

Offset: 0

Harry J. Smith, Apr 30 2009

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

			3814279.104760220592209... = 3814279 + 1/(9 + 1/(1 + 1/(1 + 1/(4 + ...)))).

Harry J. Smith, Table of n, a(n) for n = 0..20000
Eric Weisstein's World of Mathematics, Transcendental Number.
Wikipedia, Irrational number, Open questions.

Cf. A003417, A064107.

ContinuedFraction[E^E^E, 96] (* Vladimir Reshetnikov, Apr 27 2013 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(exp(exp(exp(1)))); for (n=1, 20001, write("b159825.txt", n-1, " ", x[n])); }

A073226 Decimal expansion of e^e.

Original entry on oeis.org

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

Offset: 2

Rick L. Shepherd, Jul 21 2002

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, 1 < x < e < y and (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.) Proofs of these classical results and applications of them are in Marques and Sondow (2010).
e^e = lim_{n->infinity} ((n+1)/n)^((n+1)^(n+1)/n^n), n > 0 an integer; cf. [Vernescu] wherein it is also stated that the assertions of the previous comment above were proved by Alexandru Lupas in 2006. - L. Edson Jeffery, Sep 18 2012
A weak form of Schanuel's Conjecture implies that e^e is transcendental--see Marques and Sondow (2012).

			15.15426224147926418976043027262991190552854853685613976914...

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.
Simon Plouffe, exp(E) to 2000 places
J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164.
A. Vernescu, About the use of a result of Professor Alexandru Lupas to obtain some properties in the theory of the number e, Gen. Math., Vol. 15, No. 1 (2007), 75-80.

Cf. A073233 (Pi^Pi), A049006 (i^i), A001113 (e), A073227 (e^e^e), A004002 (Benford numbers), A056072 (floor(e^e^...^e), n e's), A072364 ((1/e)^(1/e)), A030178 (limit of (1/e)^(1/e)^...^(1/e)), A073229 (e^(1/e)), A073230 ((1/e)^e).

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

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

{ default(realprecision, 20080); x=exp(1)^exp(1)/10; for (n=2, 20000, d=floor(x); x=(x-d)*10; write("b073226.txt", n, " ", d)); } \\ Harry J. Smith, Apr 30 2009

Equals Sum_{n>=0} e^n/n!. - Richard R. Forberg, Dec 29 2013

Equals Product_{n>=0} e^(1/n!). - Amiram Eldar, Jun 29 2020

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

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

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

Original entry on oeis.org

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

A073232 Decimal expansion of ((1/e)^(1/e))^(1/e).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 22 2002

			0.87342301849311664298903234866...

Cf. A001113 (e), A068985 (1/e), A072364 ((1/e)^(1/e)), A073231 ((1/e)^(1/e)^(1/e)), A073228 ((e^e)^e), A073227 (e^e^e).

RealDigits[((1/E)^(1/E))^(1/E),10,120][[1]]  (* Harvey P. Dale, Apr 21 2011 *)

PARI
```
exp(-exp(-2))
```

Equals e^(-e^(-2)).

A073231 Decimal expansion of (1/e)^(1/e)^(1/e).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 22 2002

			0.50047350056363684054513490133...

Cf. A001113 (e), A068985 (1/e), A072364 ((1/e)^(1/e)), A030178 (limit of (1/e)^(1/e)^...^(1/e)), A073232 (((1/e)^(1/e))^(1/e)), A073227 (e^e^e).

With[{c=1/E},RealDigits[c^c^c,10,120][[1]]] (* Harvey P. Dale, Jul 16 2025 *)

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

A187079 Decimal expansion of (sqrt(2 + e^e)/e)^e.

Original entry on oeis.org

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

Offset: 1

Arkadiusz Wesolowski, Mar 08 2011

(sqrt(2 + e^e)/e)^e is an approximation to Pi that's correct to five decimal digits.

			(sqrt(2+e^e)/e)^e = 3.141599009451764738125397155...

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..1000
Carlos Rivera, Puzzle 50

Cf. A000796, A073227, A073226.

SetDefaultRealField(RealField(100)); (Sqrt(2+Exp(Exp(1)))/Exp(1))^Exp(1); // G. C. Greubel, Sep 29 2018

evalf((sqrt(2+exp(1)^exp(1))/exp(1))^exp(1),120); # Muniru A Asiru, Sep 29 2018

RealDigits[N[(Sqrt[2 + E^E]/E)^E, 200]][[1]] (* Arkadiusz Wesolowski, Mar 08 2011 *)

default(realprecision, 200); e=exp(1); x=(sqrt(2+e^e)/e)^e; for(n=1, 200, d=floor(x); x=(x-d)*10; print1(d, ", ")); \\ Arkadiusz Wesolowski, Mar 08 2011

A202949 Decimal expansion of (e^e)^(e^e), where e=exp(1).

Original entry on oeis.org

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

Offset: 18

M. F. Hasler, Dec 26 2011

			776486517915808457.38262707214480111269813738740893733361098023776562998338874696481792585472289...

Cf. A073226, A073227, A073228, A085667, A181180, A073232. - M. F. Hasler, Dec 26 2011

RealDigits[#^#&/@(E^E),10,120][[1]] (* Harvey P. Dale, Aug 31 2023 *)

default(realprecision,99); t=exp(1); t=t^t; t=t^t

A073226^A073226.

A171990 Least integer a(n) for which the iterated function log, iterated n times, is defined.

Original entry on oeis.org

1, 2, 3, 16, 3814280

Offset: 1

Alexis Monnerot-Dumaine, Jan 21 2010

nonn

Log(a(1)) is defined if a(1) > 0, so a(1) = 1.
Log(log(a(2))) is defined if log(a(2)) > 0 => a(2) > 1 => a(2) = 2.
The sequence grows rapidly: a(6) = 2.33150...10^1656520, and is too large to include here.

			a(2) = 2 because log(log(2)) is defined and log(log(1)) is not;
a(3) = 3 because log(log(log(3))) is defined;
a(4) = 16 because log(log(log(log(16)))) is defined.
From _Robert G. Wilson v_, Jul 05 2022: (Start)
a(3) = ceiling(A001113).
a(4) = ceiling(A073226).
a(5) = ceiling(A073227).
a(6) = ceiling(A085667). (End)

Cf. A074785 (log(log(2))), A085667, A004002, A001113, A073226, A073227, A085667.

a(n) = my(k=1); while(1, my(s=k, i=0); while(s > 0, s=log(s); if(s > 0, i++)); if(i==n-1, return(k)); k++) \\ Felix Fröhlich, Nov 22 2015

For n > 2, a(n) = ceiling(e^(e^(...))) where e appears n-2 times.

cp's OEIS Frontend

A159825 Continued fraction for e^e^e A073227.

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A073226 Decimal expansion of e^e.

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Magma

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

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

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

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Magma

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A073232 Decimal expansion of ((1/e)^(1/e))^(1/e).

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A073231 Decimal expansion of (1/e)^(1/e)^(1/e).

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A187079 Decimal expansion of (sqrt(2 + e^e)/e)^e.

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