A004002 -id:A004002 - cp's OEIS frontend

A073226 Decimal expansion of e^e.

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

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

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

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

A073236 Pi^Pi^...^Pi (n times) rounded to nearest integer.

Original entry on oeis.org

1, 3, 36, 1340164183006357435

Offset: 0

Rick L. Shepherd, Jul 25 2002

nonn

Decimal expansions (before rounding) of Pi (A000796), Pi^Pi (A073233) and Pi^Pi^Pi (A073234) correspond to a(1), a(2) and a(3), respectively. All four terms are equivalent if floor is used instead of round. See A073237 for same sequence but using ceiling. This sequence is the analog of A004002, which deals with e.

a(4) has 666262452970848504 digits. - Mateusz Winiarski, Mar 23 2020; corrected by Martin Renner, Aug 23 2023

Cf. A000796 (Pi), A073233 (Pi^Pi), A073234 (Pi^Pi^Pi), A073237 (Ceiling of Pi^Pi^...^Pi, n times), A004002 (Benford numbers).

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

Round[NestList[Power[Pi, #] &, 1, 3]] (* Alonso del Arte, Jul 02 2014 *)

p=0; for(n=0,3, p=Pi^p; print1(round(p),",")) \\ n = 4 produces too large an exponent for PARI.

a(n) = round(Pi^Pi^...^Pi), where Pi occurs n times, a(0) = 1 (=Pi^0).

A225053 Second terms of continued fractions for power towers e, e^e, e^e^e, ...

Original entry on oeis.org

1, 6, 9, 4

Offset: 1

Vladimir Reshetnikov, Apr 25 2013

It was conjectured (but remains unproved) that none of the power towers e, e^e, e^e^e, ... are integers. If so, the corresponding continued fractions contain at least 2 terms. If the conjecture fails, let the corresponding a(n) = 0.

			a(3) = 9 because floor(1/frac(e^e^e)) = 9, since e^e^e ~ 3814279.10476.

Eric Weisstein's World of Mathematics, e.
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Tetration, Open questions

Cf. A003417, A064107, A159825, A225064, A004002.

A056072 yields the first term of the continued fraction.

$MaxExtraPrecision = Infinity; terms = 4; Map[Function[x, ContinuedFraction[x, 2][[2]]], NestList[Exp, E, terms - 1]]

A073237 a(n) = ceiling(Pi^Pi^...^Pi), where Pi appears n times.

Original entry on oeis.org

1, 4, 37, 1340164183006357436

Offset: 0

Rick L. Shepherd, Jul 25 2002

nonn

Decimal expansions (before taking ceiling) of Pi (A000796), Pi^Pi (A073233) and Pi^Pi^Pi (A073234) correspond to a(1), a(2) and a(3), respectively. See A073236 for same sequence rounded to nearest integer. This sequence is similar to A004002, which deals with e (but rounds).

a(4) has 666262452970848504 digits. - Martin Renner, Aug 19 2023

Cf. A000796 (Pi), A073233 (Pi^Pi), A073234 (Pi^Pi^Pi), A073236 (Pi^Pi^...^Pi, n times, rounded), A004002 (Benford numbers), A056072 (similar to A004002 but takes floor).

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

p=0; for(n=0,3, p=Pi^p; print1(ceil(p),",")) \\ n=4 produces too large an exponent for PARI.

a(n) = ceiling(Pi^Pi^...^Pi), where Pi occurs n times, a(0) = 1 (=Pi^0).

A056165 e^[e^[e^ ... [e^0] ... ]], n high, where [] is "floor".

Original entry on oeis.org

1, 2, 7, 1096

Offset: 0

Robert G. Wilson v, Jul 31 2000

Cf. A004002, A056072.

The next term is 9.69956295034023667235191839... *10^475

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

A073226 Decimal expansion of e^e.

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A073233 Decimal expansion of Pi^Pi.

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A073227 Decimal expansion of e^e^e.

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

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A073236 Pi^Pi^...^Pi (n times) rounded to nearest integer.

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A225053 Second terms of continued fractions for power towers e, e^e, e^e^e, ...

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A073237 a(n) = ceiling(Pi^Pi^...^Pi), where Pi appears n times.

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