Let $\tau(n)$ count the number of divisors of $n$. Is there some $n > 24$ such that $$\max_{m < n}(m + \tau(m)) \leq n + 2?$$

Erdős says 'it is extremely doubtful' that there are infinitely many such $n$, and in fact suggests that $$lim_{n\to\infty} \max_{m < n}(\tau(m) + m − n) = \infty.$$