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 \[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\]

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