Problem Statement
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 $n$?
Categories:
Number Theory
Progress
Asked by Erdős at Oberwolfach in 1986. 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]. This is not obvious; details are given in a paper of Erdős and Ivić (see page 69 of [ErIv90]).Schinzel and Wirsing [ScWi87] have proved $F(n) \gg \log n$.
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 $n$).
Source: erdosproblems.com/1106 | Last verified: January 19, 2026