Problem 1106

Erdős Problem #1106: declared status 'open'. Formalized: yes. Let $p(n)$ denote the partition function of $n$ and let $F(n)$ count the number of distinct prime factors of$$\prod_{1\leq k\leq n}p(k).$$Does $F(n)\to \infty$ with $n$? Is $F(n)>n$ for all sufficiently large $n$? Current best: Schinzel noted in the Oberwolfach problem book that $F(n)\to \infty$ follows from the asymptotic formula for $p(n)$ and a result of Tijdeman [Ti73]. Ono [On00] has proved that every prime divides $p(n)$ for some $n\geq 1$ (indeed this holds, for any fixed prime, for a positive density set of $n$). Prize: no. OEIS: A194259, A194260. Tags: number theory.