OPEN

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

A problem of Erdős and Selfridge. This is true for $n=24$. The $n+2$ is best possible here since
\[\max(\tau(n-1)+n-1,\tau(n-2)+n-2)\geq n+2.\]

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].