I prove a lemma (I take from a book ) Consider an inject function $ f[a,b]\to [c,d]$ and decrease. $ \sum_{a\leq k\leq b}f(k) - \sum_{c\leq k\leq d}f^{ - 1}(k) = [b]S(c) - [d]S(a)$ $ S(x) = [x]$ if $ x\not\in Z$ $ S(x) = 0$ if $ x = 0$ $ S(x) = x - 1$ if $ x\in Z$ and different from 0. Now consider the function : $ f[1,n]\to\ [n,n^2]$ $ f(x) = \frac {n^2}{x}$ then $ f(x)$ is inject and decrease. So $ \sum_{k = 1}^n[\frac {n^2}{k} - \sum_{k = n}^{n^2}[\frac {n^2}{k}] = n(n - 1)$ $ \Rightarrow \sum_{k = 2}^n[\frac {n^2}{k}] = \sum_{k = n + 1}^{n^2}[\frac {n^2}{k}]$

I think that book by Titu Andreescu.