A258500 Decimal expansion of the nontrivial real solution of x^(3/2) = (3/2)^x.

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

Offset

1,1

Links

Table of n, a(n) for n=1..101.

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

Formula

x0 = -((x*ProductLog(-1, -(log(x)/x)))/log(x)), with x = 3/2, where ProductLog is the Lambert W function.

Example

x0 = 7.408764686965774521957295028510614389804171141074...
z = x0^(3/2) = 20.16595073003535058942970947434890012034363496 ...
z > e^e = 15.15426224... = A073226.

Mathematica

x0 = -((x*ProductLog[-1, -(Log[x]/x)])/Log[x]) /. x -> 3/2; RealDigits[x0, 10, 101] // First
RealDigits[x/.FindRoot[x^(3/2)==(3/2)^x, {x, 7}, WorkingPrecision->120], 10, 120][[1]] (* Harvey P. Dale, Dec 07 2024 *)

Crossrefs

Cf. A073226, A194556, A194557, A258501 (x^(5/2)=(5/2)^x), A258502 (x^(7/2)=(7/2)^x).

Sequence in context: A199955 A196624 A157413 * A243308 A293609 A294514

Adjacent sequences: A258497 A258498 A258499 * A258501 A258502 A258503

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Jun 01 2015

Status

approved