jmc

Consider a sequence $x_1,$ $x_2,$ $x_3,$ $\dots$ defined by $$\begin{array}{rl}{x}_1& =\sqrt[3]{3},\\ x_2& =(\sqrt[3]{3}{)}^{\sqrt[3]{3}},\end{array}$$and in general, $$x_n=(x_{n-1}{)}^{\sqrt[3]{3}}$$for $n>1.$ What is the smallest value of $n$ for which $x_n$ is an integer?