Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the 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