ILL 1992 Problem 43

Find the number of positive integers $n$ satisfying $\phi(n) | n$ such that $$\sum_{m=1}^{\infty} \left( \left[ \frac nm \right] - \left[\frac{n-1}{m} \right] \right) = 1992$$ What is the largest number among them? As usual, $\phi(n)$ is the number of positive integers less than or equal to $n$ and relatively prime to $n.$