A258502 - OEIS

%I #17 Apr 19 2019 10:02:59

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

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

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

%N Decimal expansion of the nontrivial real solution of x^(7/2) = (7/2)^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(-(log(x)/x)))/log(x)), with x = 7/2, where ProductLog is the Lambert W function.

%e x0 = 2.189697551175613504808316814457313054952031983651039793...

%e z = x0^(7/2) = 15.53618787439250843837688346448101455506861788472622...

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

%t x0 = -((x*ProductLog[-(Log[x]/x)])/Log[x]) /. x -> 7/2; RealDigits[x0, 10, 101] // First

%t RealDigits[x/.FindRoot[x^(7/2)==(7/2)^x,{x,2},WorkingPrecision-> 120]][[1]] (* _Harvey P. Dale_, Apr 19 2019 *)

%Y Cf. A073226, A194556, A194557, A258500 (x^(3/2)=(3/2)^x), A258501 (x^(5/2)=(5/2)^x).

%K nonn,cons,easy

%O 1,1

%A _Jean-François Alcover_, Jun 01 2015