Let $\alpha$ and $\beta$ be the roots of $x^2 - \sqrt{6}x + 3 = 0$. If $\alpha^n + \beta^n$ is an integer for $n \geq 1$, then the greatest value of $n$ for which $\alpha^n + \beta^n$ is NOT an integer is $\_\_\_\_$.
Let $\alpha$ and $\beta$ be the roots of $x^2 - \sqrt{6}x + 3 = 0$. If $\alpha^n + \beta^n$ is an integer for $n \geq 1$, then the greatest value of $n$ for which $\alpha^n + \beta^n$ is NOT an integer is $\_\_\_\_$.