A194556 - OEIS

%I #18 Aug 21 2023 11:22:57

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

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

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

%N Decimal expansion of (9/4)^(27/8) = (27/8)^(9/4).

%C 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 = 2 gives (x,y) = (9/4,27/8). See Sondow and Marques 2010, pp. 155-157.

%C (9/4)^(27/8) = (27/8)^(9/4) corresponds to (4/9)^(4/9) = (8/27)^(8/27) (see A194789) under the equivalence x^y = y^x <==> (1/x)^(1/x) = (1/y)^(1/y).

%H J. Sondow and D. Marques, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_37_from151to164.pdf">Algebraic and transcendental solutions of some exponential equations</a>, Annales Mathematicae et Informaticae, 37 (2010), 151-164.

%H <a href="/index/Al#algebraic_04">Index entries for algebraic numbers, degree 4</a>

%F -((9*ProductLog(-1, -(4/9)*log(9/4)))/(4*log(9/4))), where ProductLog is the Lambert W function, simplifies to 27/8. - _Jean-François Alcover_, Jun 01 2015

%e 15.438887358552583183604460013074909718871494279680272412854330453294418363...

%t RealDigits[ (9/4)^(27/8), 10, 100] // First

%Y Cf. A073226 (e^e), A194557 (sqrt(3)^sqrt(27) = sqrt(27)^sqrt(3)), A194789 ((4/9)^(4/9) = (8/27)^(8/27)).

%K nonn,cons

%O 2,2

%A _Jonathan Sondow_, Aug 30 2011