Question 14
For each integer $n \ge 2,$ let $A(n)$ be the area of the region in the coordinate plane defined by the inequalities $1 \le x \lt n$ and $0 \le y \le x\lfloor\sqrt{x}\rfloor,$ where $\lfloor\sqrt{x}\rfloor$ is the greatest integer not exceeding $\sqrt{x}.$ Find the number of values of $n$ with $2 \le n \le 1000$ for which $A(n)$ is an integer.