CN-SMO 2007 Problem 3

Let $a_i=min\{ k+\dfrac{i}{k}|k \in N^*\}$ , determine the value of $S_{n^2}=[a_1]+[a_2]+\cdots +[a_{n^2}]$ , where $n\ge 2$ . ( $[x]$ denotes the greatest integer not exceeding x)