A073243 -id:A073243 - cp's OEIS frontend

A073233 Decimal expansion of Pi^Pi.

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

A030437 Decimal expansion of x such that x^x = Pi.

Original entry on oeis.org

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

Offset: 1

James L. Dean (csvcjld(AT)nomvs.lsumc.edu)

			x = 1.8541059679210264327483707184102932454292... .

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000796 (Pi), A100947 (continued fraction), A073243 (reciprocal).

x^x=Pi; solve(%,x); evalf(%, 140); # solution is log(Pi)/LambertW(log(Pi)), where LambertW is the Omega function.

x=Pi; RealDigits[Log[x]/ProductLog[Log[x]],10,6! ][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2010 *)
RealDigits[x/.FindRoot[x^x==Pi,{x,1},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Nov 29 2024 *)

solve(x=1, 2, x^x-Pi) \\ Michel Marcus, Jan 14 2015

exp(lambertw(log(Pi))) \\ Charles R Greathouse IV, Nov 11 2017

More terms from Simon Plouffe

Better name from Jon E. Schoenfield, Dec 30 2014

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)
```

A073241 Decimal expansion of (1/Pi)^(1/Pi)^(1/Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 27 2002

			0.45150834553657280522199381804...

Cf. A000796 (Pi), A049541 (1/Pi), A073240 ((1/Pi)^(1/Pi)), A073243 (limit of (1/Pi)^(1/Pi)^...^(1/Pi)), A073242 (((1/Pi)^(1/Pi))^(1/Pi)), A073234 (Pi^Pi^Pi).

With[{c=1/Pi},RealDigits[c^c^c,10,120][[1]]] (* Harvey P. Dale, Mar 10 2015 *)

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

A293009 Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/Pi.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Mar 16 2018

			0.56501844596024150528994096062245192028392680078511838285519...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Tetration

Cf. A000796, A049541, A073243, A277522, A277651, A300916.

RealDigits[Pi*Exp[-2*LambertW[Log[Pi]]]/(1+LambertW[Log[Pi]]), 10, 100][[1]] (* G. C. Greubel, Sep 09 2018 *)

Pi*exp(-2*lambertw(log(Pi)))/(1+lambertw(log(Pi))) \\ Michel Marcus, Mar 16 2018

Equals Pi*exp(-2*LambertW(log(Pi)))/(1+LambertW(log(Pi))).

A100120 Limit of the power tower t(2)^(t(3)^(t(5)^(t(7)^(...)))) with t(n)=n!^(1/n!) and n taking prime values.

Original entry on oeis.org

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

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 09 2004

Let f(n)=n!^(1/n!). Then this number is f(2)^(f(3)^(f(5)^(...)))

			1.604651218410055827983866511015173398808649754699580...

Cf. A077589, A077590, A073243, A098454, A098584.

default(realprecision,100):t=1:forstep(n=100,1,-1,t=(prime(n)!^(1/prime(n)!))^t):return(t)

Offset corrected by R. J. Mathar, Feb 05 2009

A194346 Decimal expansion of h_o(1/17), where h_o(x) is the odd infinite power tower function.

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Aug 27 2011

The odd infinite power tower function is h_o(x) = lim f(n,x) as n --> infinity, where f(n+1,x) = x^x^(f(n,x)) and f(1,x) = x. The even infinite power tower function h_e(x) is the same limit except with f(1,x) = x^x (see A194347). The limits exist if and only if 0 < x <= e^(1/e). If (1/e)^e <= x <= e^(1/e), then h_o(x) = h_e(x) = h(x) (the infinite power tower function-see the comments in A073230) and y = h(x) is a solution of x^y = y. If 0 < x < (1/e)^e, then h_o(x) < h_e(x), and two solutions of x^x^y = y are y = h_o(x) and y = h_e(x). For example, y = h_o(1/16) = 1/4 and y = h_e(1/16) = 1/2 are solutions of (1/16)^(1/16)^y = y.

h_o(1/17) and h_e(1/17) are irrational, and at least one of them is transcendental (see Sondow and Marques 2010).

			0.204274736665518499175698745186446957991668690348422572736592466759324966133336...

See the References in Sondow and Marques 2010.

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 Definition 4.3, Figure 7, and top of p. 163.

Cf. A073229, A073230, A073243, A194347.

a = N[1/17, 100]; Do[a = (1/17)^(1/17)^a, {3000}]; RealDigits[a, 10, 100] // First
RealDigits[ Fold[ N[#2^#1, 128] &, 1/17, Table[1/17, {5710}]], 10, 105][[1]] (* Robert G. Wilson v, Mar 20 2012 *)

solve(x=0,1,17^(-17^-x)-x) \\ Charles R Greathouse IV, Mar 20 2012

A194347 Decimal expansion of h_e(1/17), where h_e(x) is the even infinite power tower function.

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Aug 27 2011

See the Comments, References and Links in A194346.

			0.560596485316498211176108570378470772301099831523122840744442447202319227572291...

Cf. A073229, A073230, A073243, A194346.

a = N[(1/17)^(1/17), 100]; Do[a = (1/17)^(1/17)^a, {3000}]; RealDigits[a, 10, 100] // First

A266092 Decimal expansion of the power tower of 1/sqrt(3): the real solution to 3^(x/2)*x = 1.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

			(1/sqrt(3))^(1/sqrt(3))^(1/sqrt(3))^(1/sqrt(3))^… = 0.686026724536251319713006846182…

Eric Weisstein's World of Mathematics, Power Tower
Eric Weisstein's World of Mathematics, Lambert W-Function

Cf. A020760, A030178, A073084, A073243, A231096.

RealDigits[(2 ProductLog[Log[3]/2])/Log[3], 10, 120][[1]]

t=log(3)/2; lambertw(t)/t \\ Charles R Greathouse IV, Apr 18 2016

Equals 2*LambertW(log(3)/2)/log(3).

A276635 Decimal expansion of the power tower of 1/(2Pi): the real solution to (2Pi)^x*x = 1.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Sep 08 2016

			(1/(2*Pi))^(1/(2*Pi))^(1/(2*Pi))^... = 0.443001457438838056674419269992719...

Eric Weisstein's World of Mathematics, Plouffe's Constants
Eric Weisstein's World of Mathematics, Power Tower
Eric Weisstein's World of Mathematics, Lambert W-Function

Cf. A019692, A073243, A061444, A086201.

RealDigits[ProductLog[Log[2 Pi]]/Log[2 Pi], 10, 120][[1]]

Equals LambertW(log(2*Pi))/log(2*Pi).

Equals exp(-LambertW(A061444)).

cp's OEIS Frontend

A073233 Decimal expansion of Pi^Pi.

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A030437 Decimal expansion of x such that x^x = Pi.

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

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A073241 Decimal expansion of (1/Pi)^(1/Pi)^(1/Pi).

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A293009 Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/Pi.

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A100120 Limit of the power tower t(2)^(t(3)^(t(5)^(t(7)^(...)))) with t(n)=n!^(1/n!) and n taking prime values.

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A194346 Decimal expansion of h_o(1/17), where h_o(x) is the odd infinite power tower function.

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PARI

A194347 Decimal expansion of h_e(1/17), where h_e(x) is the even infinite power tower function.

A276635 Decimal expansion of the power tower of 1/(2Pi): the real solution to (2Pi)^x*x = 1.