A116186 -id:A116186 - 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

A116191 Decimal expansion of imaginary part of i^(i^i), that is, Im(i^(i^i)).

Original entry on oeis.org

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

Offset: 0

Peter C. Heinig (algorithms(AT)gmx.de), Apr 15 2007

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

			i^(i^i) = 0.947158998072378380653475352018 + 0.320764449979308534660116845875 i.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 144.
Steven R. Finch, Errata and Addenda to Mathematical Constants, Jun 23 2012, Section 1.1
D. Marques and J. Sondow, Schanuel's conjecture and algebraic powers z^w and w^z with z and w transcendental, East-West J. Math., 12 (2010), 75-84.
Wikipedia, Schanuel's conjecture

Cf. A049006, A116186.

C := ComplexField(100);  Im(I^I^I); // G. C. Greubel, May 11 2019

Maple

c := sin((Pi/2)/exp(Pi/2)): Digits := 110: evalf(c, Digits)*10^105: ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Oct 23 2024

Mathematica

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

PARI

imag(I^I^I) \\ Charles R Greathouse IV, May 15 2013

Sage

numerical_approx((i^i^i).imag(), digits=100) # G. C. Greubel, May 11 2019

Formula

Equals sin((Pi/2)/exp(Pi/2)). - Peter Luschny, Oct 23 2024

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A073233 Decimal expansion of Pi^Pi.

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A116191 Decimal expansion of imaginary part of i^(i^i), that is, Im(i^(i^i)).

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Maple

Mathematica

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