A258502 Decimal expansion of the nontrivial real solution of x^(7/2) = (7/2)^x.

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, 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, 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

Offset: 1

Author

Jean-François Alcover, Jun 01 2015

Keywords

• nonn
• cons
• easy

Examples

			x0 = 2.189697551175613504808316814457313054952031983651039793...
z = x0^(7/2) = 15.53618787439250843837688346448101455506861788472622...
z > e^e = 15.15426224... = A073226.
		

Links

• Jonathan Sondow, Diego Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164.

Crossrefs

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

Programs

Mathematica

x0 = -((x*ProductLog[-(Log[x]/x)])/Log[x]) /. x -> 7/2; RealDigits[x0, 10, 101] // First
RealDigits[x/.FindRoot[x^(7/2)==(7/2)^x,{x,2},WorkingPrecision-> 120]][[1]] (* Harvey P. Dale, Apr 19 2019 *)

Formula

x0 = -((x*ProductLog(-(log(x)/x)))/log(x)), with x = 7/2, where ProductLog is the Lambert W function.