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Claim. A. Schinzel and E. Wirsing, Multiplicative properties of the partition function, Proc. Indian Acad. Sci. Math. Sci. 97 (1987), nos. 1–3, 297–303. Write $m(N,N+R)$ for the number of multiplicatively independent values of $p(n)$ in $N\le n<N+R$, and $m(N)$ for that number in $1\le n\le N$. The paper's Theorem states that there is an $N_0$ such that

$$m(N,N+R)\ge R\,\frac{\log N-\log R}{\tfrac32\log N+R\log2}$$

for $N\ge N_0$ and all $R\in\mathbb N$, and that the same lower bound applies to the number of distinct prime factors of $\prod_{N\le n<N+R}p(n)$. Its Corollary 2 gives $m(N,N+R)\ge(1/\log2-o(1))\log N$ when $R/\log N\to\infty$, and the paper then states that

$$\omega\Bigl(\prod_{n=1}^{N}p(n)\Bigr)\ge m(N)\ge(1-\varepsilon)\frac{\log N}{\log2} \qquad(N\ge N_0(\varepsilon)).$$

The left side is $F(N)$, the number of distinct prime factors of $p(1)p(2)\cdots p(N)$ in Problem 1106, so $F(n)\ge(1-\varepsilon)\log n/\log2$ for all large $n$, and in particular $F(n)\gg\log n$; the site's commentary and the formal-conjectures entry both credit that bound to this paper. The inequality $\omega\ge m$ is elementary: positive integers built from $s$ primes lie in a free abelian group of rank $s$, so $r$ multiplicatively independent values force at least $r$ distinct primes. Ono [On00] cites the paper for the bound $\gg\log\log X$ on the number of primes $m<X$ that divide some $p(n)$.

Covers. The first question: $F(n)\to\infty$, at the rate $F(n)\ge(1-\varepsilon)\log n/\log2$. The second question, whether $F(n)>n$ for all large $n$, is not addressed.

Depends on. No page of this wiki: the bound on $F(n)$ is stated in the paper.

Acceptance. Refereed: the paper appeared in Proceedings of the Indian Academy of Sciences (Mathematical Sciences) in December 1987. The site's commentary credits $F(n)\gg\log n$ to this paper, but the site labels the problem OPEN, so the commentary is not acceptance and no reviewed is listed. The formal-conjectures file states the first question as erdos_1106.parts.i with answer(True), the category research solved and a sorry body, citing this paper; it is a statement, not a proof.