2015 AIME I Problem 14

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 \right\rfloor, where \left\lfloor \sqrt x \right\rfloor 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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.