A194557 Decimal expansion of sqrt(3)^sqrt(27) = sqrt(27)^sqrt(3).

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

Offset: 2

Author

Jonathan Sondow, Aug 30 2011

Keywords

• nonn
• cons

Comments

Positive real numbers x < y with x^y = y^x are parameterized by (x,y) = ((1 + 1/t)^t,(1 + 1/t)^(t+1)) for t > 0. For example, t = 1/2 gives (x,y) = (sqrt(3),sqrt(27)). See Sondow and Marques 2010, pp. 155-157.

Examples

			17.361905250953135215415714826833267582295532184890864078454696057446763745...
		

Links

• J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae, 37 (2010), 151-164.
• Index entries for transcendental numbers

Crossrefs

Cf. A073226 (decimal expansion of e^e), A194556 (decimal expansion of (9/4)^(27/8) = (27/8)^(9/4)).

Programs

Mathematica

RealDigits[ Sqrt[3]^Sqrt[27], 10, 100] // First

Formula

-2*sqrt(3)*ProductLog(-1, -log(3)/(2*sqrt(3)))/log(3), where ProductLog is the Lambert W function, simplifies to sqrt(27). - Jean-François Alcover, Jun 01 2015