How many positive integers greater than $2$ and less than $2023$ are there such that $\frac{1}{\sqrt{2+2\sqrt{2}+1}} + \frac{1}{\sqrt{4+2\sqrt{6}+1}} + \frac{1}{\sqrt{6+2\sqrt{12}+1}} + \dots + \frac{1}{\sqrt{2n+2\sqrt{n^2+n}+1}}$ is an integer?