What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer? $\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be \[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\]