In [Er79] Erdős says 'it is extremely doubtful' that there are infinitely many such $n$, and in fact suggets that \[\lim_{n\to \infty}\max_{m<n}(\tau(m)+m-n)=\infty.\]
In [Er79d] Erdős says it 'seems certain' that for every $k$ there are infinitely many $n$ for which \[\max_{n-k<m<n}(m+\tau(m))\leq n+2,\] but 'this is hopeless with our present methods', although it follows from Schinzel's Hypothesis H.
See also [413].