Problem 14 (2015 AIME I)

For each integer $n \ge 2$, let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1\le x \le n$ and 0\le y \le x \left\lfloor \sqrt x ight floor, where \left\lfloor \sqrt x ight floor is the greatest integer not exceeding $\sqrt x$. Find the number of values of $n$ with $2\le n \le 1000$ for which $A(n)$ is an integer.