Problem 413

Erdős Problem #413: declared status 'open'. Formalized: yes. Let $\omega(n)$ count the number of distinct primes dividing $n$. Are there infinitely many $n$ such that, for all $m<n$, we have $m+\omega(m) \leq n$? Can one show that there exists an $\epsilon>0$ such that there are infinitely many $n$ where $m+\epsilon \omega(m)\leq n$ for all $m<n$? Prize: no. OEIS: A005236. Tags: iterated functions, number theory.