A258501 - OEIS

%I #10 Dec 10 2016 19:37:58

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

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

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

%N Decimal expansion of the nontrivial real solution of x^(5/2) = (5/2)^x.

%H Jonathan Sondow, Diego Marques, <a href="http://arxiv.org/abs/1108.6096">Algebraic and transcendental solutions of some exponential equations</a>, Annales Mathematicae et Informaticae 37 (2010) 151-164.

%F x0 = -((x*ProductLog(-1, -(log(x)/x)))/log(x)), with x = 5/2, where ProductLog is the Lambert W function.

%e x0 = 2.97028705025575877937998429103168637323950439632715025453459...

%e z = x0^(5/2) = 15.20533715980107653442006557792026842686895921352582...

%e z > e^e = 15.15426224... = A073226.

%t x0 = -((x*ProductLog[-1, -(Log[x]/x)])/Log[x]) /. x -> 5/2; RealDigits[x0, 10, 102] // First

%Y Cf. A073226, A194556, A194557, A258500 (x^(3/2)=(3/2)^x), A258502 (x^(7/2)=(7/2)^x).

%K nonn,cons,easy

%O 1,1

%A _Jean-François Alcover_, Jun 01 2015