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

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

0,1

Comments

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

Links

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

Formula

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

Example

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

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

Prog

(PARI) imag(I^I^I) \\ Charles R Greathouse IV, May 15 2013
(Magma) C<I> := ComplexField(100);  Im(I^I^I); // G. C. Greubel, May 11 2019
(Sage) numerical_approx((i^i^i).imag(), digits=100) # G. C. Greubel, May 11 2019

Crossrefs

Cf. A049006, A116186.

Sequence in context: A010604 A067585 A173787 * A257303 A048199 A335998

Adjacent sequences: A116188 A116189 A116190 * A116192 A116193 A116194

Keyword

nonn,cons

Author

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

Status

approved