A194555 -id:A194555 - 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

A194348 Decimal expansion of sqrt(2)^sqrt(2)^sqrt(2).

Original entry on oeis.org

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

Offset: 1

Jonathan Sondow, Aug 28 2011

If Schanuel's Conjecture is true, then sqrt(2)^sqrt(2)^sqrt(2) is transcendental (see Marques and Sondow 2010, p. 79).

			1.76083955588002809075664989563837274807980409431850990464638822505342...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Steven Finch, Errata and Addenda to Mathematical Constants, Jun 23 2012, Section 1.1
D. Marques and Jonathan Sondow, Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental, arXiv:1010.6216 [math.NT], 2010-2011; East-West J. Math., 12 (2010), 75-84.
Wikipedia, Schanuel's conjecture

Cf. A002193 (decimal expansion of sqrt(2)), A078333 (decimal expansion of sqrt(2)^sqrt(2)), A194555 (the decimal expansion of the real part of I^e^Pi).

SetDefaultRealField(RealField(100)); Sqrt(2)^Sqrt(2)^Sqrt(2); // G. C. Greubel, Aug 19 2018

RealDigits[ Sqrt[2]^Sqrt[2]^Sqrt[2], 10, 100] // First

sqrt(2)^sqrt(2)^sqrt(2) \\ Charles R Greathouse IV, May 14 2014

(x->x^x^x)(sqrt(2)) \\ Charles R Greathouse IV, May 14 2014

A194554 Decimal expansion of the absolute value of the imaginary part of i^(e^Pi), where i = sqrt(-1).

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow, Aug 28 2011

If Schanuel's Conjecture is true, then i^e^Pi is transcendental (see Marques and Sondow 2010, p. 79).

			i^e^Pi = 0.2192048949... - 0.9756788478...*i

G. C. Greubel, Table of n, a(n) for n = 0..10000
Steven Finch, Errata and Addenda to Mathematical Constants, Jun 23 2012, Section 1.1
D. Marques and Jonathan Sondow, Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental, https://arxiv.org/abs/1010.6216 [math.NT], 2010-2011; East-West J. Math., 12 (2010), 75-84.
Wikipedia, Schanuel's conjecture

Cf. A039661 (decimal expansion of e^Pi), A194555 (real part).

RealDigits[Im[I^E^Pi], 10, 100] // First

abs(imag(I^(exp(Pi)))) \\ Michel Marcus, Aug 19 2018

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A073233 Decimal expansion of Pi^Pi.

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A194348 Decimal expansion of sqrt(2)^sqrt(2)^sqrt(2).

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A194554 Decimal expansion of the absolute value of the imaginary part of i^(e^Pi), where i = sqrt(-1).

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Mathematica

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