A073233 -id:A073233 - cp's OEIS frontend

A159824 Continued fraction for Pi^Pi (cf. A073233).

36, 2, 6, 9, 2, 1, 2, 5, 1, 1, 6, 2, 1, 291, 1, 38, 50, 1, 2, 5, 4, 1, 2, 2, 1, 5, 1, 4, 13, 2, 1, 4, 3, 3, 1, 2, 25, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 1, 43, 1, 2, 7, 3, 1, 1, 1, 2, 4, 2, 1, 1, 3, 1, 3, 3, 2, 2, 16, 3, 5, 2, 1, 5, 2, 1, 10, 1, 1, 3, 1, 13, 1, 1, 3, 1, 10, 4, 1, 1, 1, 38, 1, 2, 2, 1, 1, 3

Offset: 0

Harry J. Smith, Apr 30 2009

nonn

			36.4621596072079117709908260... = 36 + 1/(2 + 1/(6 + 1/(9 + 1/(2 + ...)))).

Harry J. Smith, Table of n, a(n) for n = 0..20000

Cf. A001076, A001203, A001204, A002945, A010767, A011002, A011003, A073233, A073238, A179613, A179615, A179616, A179617.

ContinuedFraction[Pi^Pi,200] (* Vladimir Joseph Stephan Orlovsky, Jul 20 2010 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(Pi^Pi); for (n=1, 20001, write("b159824.txt", n-1, " ", x[n])); }

Edited by N. J. A. Sloane, Jul 22 2010

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

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)
```

A202955 Decimal expansion of Pi^Pi^Pi^Pi.

Original entry on oeis.org

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

Offset: 666262452970848504

M. F. Hasler, Dec 26 2011

The offset equals the floor(Pi^Pi^Pi*Log_10(Pi))+1. - Robert G. Wilson v, Mar 13 2014

Whether this number is an integer or not is an open question. It is also an open question whether Pi^Pi^Pi^...^Pi^Pi n times is an integer for any natural n > 4. - Eliora Ben-Gurion, Nov 17 2019

			9.080222455390617769723931713284287746516046358131897359946935926336845199... *10^666262452970848503

Cf. A073236, A085667, A000796 (Pi), A073233 (Pi^Pi), A073234 (Pi^Pi^Pi), A073235 ((Pi^Pi)^Pi), A202953 ((Pi^Pi)^(Pi^Pi)).

nbrdgt = 105; f[base_, exp_] := RealDigits[ 10^FractionalPart[ N[exp*Log10[base], nbrdgt + Floor[Log10[exp]] + 2]], 10, nbrdgt][[1]]; f[Pi, Pi^Pi^Pi] (* Robert G. Wilson v, Mar 13 2014 *)

LP(a,b)=[10^frac(a=log(a)/log(10)*b),a\1] /* returns [m,e] such that a^b = m*10^e */
LP(Pi,Pi^Pi^Pi)

A000796^A073234.

A073234 Decimal expansion of Pi^(Pi^Pi).

Original entry on oeis.org

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

Offset: 19

Rick L. Shepherd, Jul 25 2002

			1340164183006357435.29744912964...

Christopher Creutzig & Walter Oevel, MuPAD Tutorial. Berlin: Springer-Verlag (2004): 339.

Harry J. Smith, Table of n, a(n) for n = 19..20000

Cf. A000796 (Pi), A073233 (Pi^Pi), A202955 (Pi^Pi^Pi^Pi), A073236 (Pi^Pi^...^Pi, n times, rounded), A073235 ((Pi^Pi)^Pi), A073241 ((1/Pi)^(1/Pi)^(1/Pi)).

RealDigits[Pi^Pi^Pi, 10, 100][[1]] (* Alonso del Arte, Jul 03 2012 *)

DIGITS := 100:
float(PI^(PI^PI)) // from Creutzig & Oevel (2004)

PARI
```
Pi^Pi^Pi
```

{ default(realprecision, 20100); x=Pi^Pi^Pi/10^18; for (n=19, 20000, d=floor(x); x=(x-d)*10; write("b073234.txt", n, " ", d)); } \\ Harry J. Smith, May 01 2009, corrected May 19 2009, had -n

A073238 Decimal expansion of Pi^(1/Pi).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jul 25 2002

Pi^(1/Pi) = 1/((1/Pi)^(1/Pi)) (reciprocal of A073240).

			1.43961949584759068833649080497...

Cf. A000796 (Pi), A049541 (1/Pi), A073239 ((1/Pi)^Pi), A073240 ((1/Pi)^(1/Pi)), A073233 (Pi^Pi).

RealDigits[N[Pi^(1/Pi),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 27 2012 *)

PARI
```
Pi^(1/Pi)
```

A073240 Decimal expansion of (1/Pi)^(1/Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 27 2002

(1/Pi)^(1/Pi) = Pi^(-1/Pi) = 1/(Pi^(1/Pi)) (reciprocal of A073238).

			0.69462799224682615312438376141...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A000796 (Pi), A049541 (1/Pi), A073241 ((1/Pi)^(1/Pi)^(1/Pi)), A073243 (limit of (1/Pi)^(1/Pi)^...^(1/Pi)), A073238 (Pi^(1/Pi)), A073239 ((1/Pi)^Pi), A073233 (Pi^Pi).

First[RealDigits[(1/Pi)^(1/Pi),10,100]] (* Paolo Xausa, Nov 07 2023 *)

PARI
```
(1/Pi)^(1/Pi)
```

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).

A231736 Decimal expansion of the natural logarithm of Pi^Pi.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Nov 13 2013

			3.59627499972915819808600175164636038136917928975387723049724412082...

Stanislav Sykora, Table of n, a(n) for n = 1..5000

Cf. A000796, A053510, A073233, A231737.

RealDigits[Pi * Log[Pi], 10, 120][[1]] (* Amiram Eldar, May 17 2023 *)

PARI
```
Pi*log(Pi)
```

Equals Pi*log(Pi).

A073235 Decimal expansion of (Pi^Pi)^Pi.

Original entry on oeis.org

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

Offset: 5

Rick L. Shepherd, Jul 25 2002

			80662.6659385545967384983601897...

Cf. A000796 (Pi), A073233 (Pi^Pi), A073234 (Pi^Pi^Pi), A073241 ((1/Pi)^(1/Pi)^(1/Pi)), A073242 (((1/Pi)^(1/Pi))^(1/Pi)).

RealDigits[(Pi^Pi)^Pi, 10, 105][[1]] (* Alonso del Arte, Jul 03 2012 *)

PARI
```
(Pi^Pi)^Pi
```

cp's OEIS Frontend

A159824 Continued fraction for Pi^Pi (cf. A073233).

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A073226 Decimal expansion of e^e.

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A073244 Decimal expansion of Pi - e.

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A202955 Decimal expansion of Pi^Pi^Pi^Pi.

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A073234 Decimal expansion of Pi^(Pi^Pi).

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A073238 Decimal expansion of Pi^(1/Pi).

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A073240 Decimal expansion of (1/Pi)^(1/Pi).

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PARI