2015 AIME I Problem 14

For each integer n \geq 2, let A(n) be the area of the region in the coordinate plane defined by the inequalities 1 \leq x<n and 0 \leq y \leq x\lfloor\sqrt{x}\rfloor, where \lfloor\sqrt{x}\rfloor is the greatest integer not exceeding \sqrt{x}. Find the number of values of n with 2 \leq n \leq 1000 for which A(n) is an integer.