OPEN
This is open, and cannot be resolved with a finite computation.
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 $n$?
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 $n$).
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This page was last edited 16 November 2025.
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