Let $p(n)$ denote the partition function of $n$ and let $F(n)$ count the number of distinct prime factors of$$\prod _{1\le k\le n}p(k).$$Does $F(n)\to \infty$ with $n$? Is $F(n)>n$ for all sufficiently large $n$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Write [ F(n)=\omega\Big(\prod_{k=1}^n p(k)\Big), ] where (\omega(\cdot)) is the number of **distinct** prime divisors. Equivalently, $F(n)$ is the size of the union of the prime-divisor sets of (p(1),p(2),\dots,p(n)). In particular $F(n)$ is **nondecreasing** in $n$.

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.