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Claim. Corollary 2 of K. Ono, Distribution of the partition function modulo $m$, Ann. of Math. (2) 151 (2000), no. 1, 293–307, digested on the card [[../library/arithmetic_functions/ono_2000_distribution_partition_function_modulo_m/_index|Ono 2000]]: Erdős's conjecture that every prime $m$ divides $p(n)$ for some $n$ is true. The corollary adds counts of the $n\le X$ with $p(n)\equiv0\pmod m$: $\gg_m X$ for primes $m\ge5$, from Theorem 1, and $\gg_m\sqrt X$ for $m=2$; for $m=3$ it rests on $p(3)=3$, and the paper notes that it is not known whether $p(n)\equiv0\pmod3$ for infinitely many $n$.

The bridge to Problem 1106 is not in the paper; it is stated here. Given $k$, each of the first $k$ primes divides some $p(n_i)$, so for $n\ge\max_i n_i$ the product $p(1)p(2)\cdots p(n)$ has at least $k$ distinct prime factors; hence $F(n)\to\infty$. The corollary gives no rate.

Covers. The first question: $F(n)\to\infty$. The second question, whether $F(n)>n$ for all large $n$, is not addressed.

Depends on. No page of this wiki: the bridge is proved above.

Acceptance. Refereed: the paper appeared in the Annals of Mathematics in January 2000. The site's commentary credits this paper with the statement that every prime divides some $p(n)$, but the site labels the problem OPEN, so the commentary is not acceptance and no reviewed is listed. The page is dated by the journal issue, which precedes the arXiv posting of 17 August 2000.