A073240 -id:A073240 - 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

A073243 Decimal expansion of exp(-LambertW(log(Pi))), solution to x = 1/Pi^x.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 28 2002

Original definition: Limit of (1/Pi)^...^(1/Pi), n times, as n approaches infinity. Equals exp(-LambertW(log(Pi))).
The value can be obtained by iterating x -> 1/Pi^x with any real starting value, but convergence is linear and slow: about 5 iterations are needed for each additional decimal digit. - M. F. Hasler, Nov 01 2011
According to the Weisstein link, infinite iterated exponentiation such as used here, which is referred to both as an "infinite power tower" and "h(x)" -- with graph and other notations -- "converges iff e^(-e) <= x <= e^(1/e) as shown by Euler (1783) and Eisenstein (1844)" (citing Le Lionnais and Wells references). e^(-e) = A073230. e^(1/e) = A073229. x of interest here = 1/Pi = A049541. (1/A073243)^(1/A073243) = A030437^A030437 = Pi.
If y = h(x) = x^x^x^... converges, then by substitution y = x^y. So x^x^x^... is a solution y to the equation y^(1/y) = x. - Jonathan Sondow, Aug 27 2011
The expressions involving "..." in the above comment are misleading, since the limit is not obtained by applying additional "^x" to the previous expression, i.e., iterating "t -> t^x", but corresponds to iterations of "t -> x^t". - M. F. Hasler, Nov 01 2011

			0.53934349886230120806079568445...

Stanislav Sykora, Table of n, a(n) for n = 0..1999
J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; see the Appendix, arXiv:1108.6096.
Eric Weisstein's World of Mathematics, Power Tower

Cf. A000796 (Pi), A049541 (1/Pi), A073240 ((1/Pi)^(1/Pi)), A073241 ((1/Pi)^(1/Pi)^(1/Pi)), A030437 (reciprocal of A073243), A030178 (corresponding limit for 1/e), A030797 (reciprocal of A030178).

y /. FindRoot[y^(1/y) == 1/Pi, {y, 1}, WorkingPrecision -> 100] (* Jonathan Sondow, Aug 27 2011 *)
First[RealDigits[Exp[-ProductLog[Log[Pi]]], 10, 104]] (* Vladimir Reshetnikov, Nov 01 2011 *)

/* The program below was run with precision set to 1000 digits */ /* n is the number of iterated exponentiations performed. */ /* (n turns out to be 954 with 1E-200 specified here) */ n=0; s=1/Pi; t=1; while(abs(t-s)>1E-200, t=s; s=(1/Pi)^s; n++); print(n,",",s)

solve(x=0,1,x-1/Pi^x)  \\ M. F. Hasler, Nov 01 2011

x = LambertW(log(Pi))/log(Pi), solution to Pi^x=1/x. - M. F. Hasler, Nov 01 2011

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)
```

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)
```

A073239 Decimal expansion of (1/Pi)^Pi.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 25 2002

(1/Pi)^Pi = Pi^(-Pi) = 1/(Pi^Pi) (reciprocal of A073233).

			0.02742569312329810611955627085...

Cf. A000796 (Pi), A049541 (1/Pi), A073238 (Pi^(1/Pi)), A073240 ((1/Pi)^(1/Pi)), A073233 (Pi^Pi).

Join[{0},RealDigits[(1/Pi)^Pi,10,120][[1]]] (* Harvey P. Dale, Nov 30 2011 *)

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

A073242 Decimal expansion of ((1/Pi)^(1/Pi))^(1/Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jul 28 2002

			0.89048823592032706709152866456...

Cf. A000796 (Pi), A049541 (1/Pi), A073240 ((1/Pi)^(1/Pi)), A073241 ((1/Pi)^(1/Pi)^(1/Pi)), A073235 ((Pi^Pi)^Pi), A073234 (Pi^Pi^Pi).

RealDigits[ N[(1/Pi^(1/Pi))^(1/Pi), 110]][[1]]

\\ This shorter statement is equivalent to ((1/Pi)^(1/Pi))^(1/Pi): Pi^(-Pi^-2)

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A073233 Decimal expansion of Pi^Pi.

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A073243 Decimal expansion of exp(-LambertW(log(Pi))), solution to x = 1/Pi^x.

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

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

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A073239 Decimal expansion of (1/Pi)^Pi.

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A073242 Decimal expansion of ((1/Pi)^(1/Pi))^(1/Pi).

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