$N$ is a fixed positive integer, and $r \in (0,\frac{N}{N-1}]$ is a fixed positive real number.
Consider the following problem $$\min_k \left[1 - r \left( 1 - \frac{1}{k} \right) \right] \left( \frac{N/k + 1}{\lfloor N/k \rfloor + 1} - \frac{N/k - 1}{\lfloor N/k \rfloor} \right) + \frac{1}{k} + r\left(1-\frac{1}{k}\right),$$ where $\lfloor x \rfloor$ denotes the maximal integer which is not greater than $x$.
Using matlab to draw several figures, I guess that $\lfloor N \rfloor$ or $\lceil N \rceil$ is a solution to the above problem, but can not prove it.
Many thanks for solving this problem precisely.