A073238 -id:A073238 - 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

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)
```

A231737 Decimal expansion of the natural logarithm of Pi^(1/Pi).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 13 2013

			0.3643788396759062570495877303161624138917070390986055504746692...

Stanislav Sykora, Table of n, a(n) for n = 0..5000

Cf. A000796, A053510, A073238, A231736.

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

PARI
```
log(Pi)/Pi
```

Equals log(Pi)/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
```

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

Original entry on oeis.org

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

A179617 Continued fraction for Pi^(1/Pi).

Original entry on oeis.org

1, 2, 3, 1, 1, 1, 3, 1, 1, 3, 2, 1, 6, 3, 4, 2, 1, 14, 1, 1, 5, 2, 2, 2, 1, 3, 23, 1, 26, 23, 13, 1, 1, 1, 5, 2, 8, 12, 1, 1, 1, 3, 5, 23, 31, 7, 1, 1, 2, 5, 4, 1, 1, 6, 1, 72, 4, 1, 1, 1, 7, 2, 1, 1, 2, 49, 3, 1, 4, 2, 3, 2, 1, 1, 6, 2, 3, 3, 1, 1, 26, 2, 2, 11, 5, 3, 5, 1, 2, 1, 12, 1, 558, 1, 1, 3, 1, 76

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 20 2010

Pi^(1/Pi) = 1.43961949584759068833649080497375567869829...

Cf. A001203, A073238 (decimal expansion).

Mathematica
```
ContinuedFraction[Pi^(1/Pi),200]
```

Offset changed by Andrew Howroyd, Jul 07 2024

A348261 Decimal expansion of the nontrivial number x for which x^Pi = Pi^x.

Original entry on oeis.org

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

Offset: 1

Timothy L. Tiffin, Oct 08 2021

The x-th root of x equals the Pi-th root of Pi: x^(1/x) = Pi^(1/Pi) = A073238 = 1.43961949584759... .

Like Pi, is x also transcendental?

			2.382179087993018774555593052520878...
x^Pi = Pi^x = 15.28621734783496640312486439999472... .

Cf. A000796 (Pi), A049541 (1/Pi), A073238 (Pi^(1/Pi)), A073226 (e^e, see first comment), A231737.

evalf((t-> -LambertW(-t)/t)(log(Pi)/Pi), 120);  # Alois P. Heinz, Oct 13 2021

{a, b} = NSolve[x^Pi == Pi^x, x, WorkingPrecision -> 300]; a; RealDigits[N[x/.a, 300]][[1]]

Equals -Pi*LambertW(-log(Pi)/Pi)/log(Pi). - Alois P. Heinz, Oct 13 2021

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A073233 Decimal expansion of Pi^Pi.

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

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A231737 Decimal expansion of the natural logarithm of Pi^(1/Pi).

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

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

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A179617 Continued fraction for Pi^(1/Pi).

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A348261 Decimal expansion of the nontrivial number x for which x^Pi = Pi^x.

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