If $\tau (n)$ counts the number of divisors of $n$ then let$$F(f,n)=\frac{\tau ((n+\lfloor f(n)\rfloor )!)}{\tau (n!)}.$$Is it true that$$\underset{n\to \infty }{\lim }F((\log n{)}^C,n)=\infty$$for large $C$? Is it true that $F(\log n,n)$ is everywhere dense in $(1,\infty )$? More generally, if $f(n)\le \log n$ is a monotonic function such that $f(n)\to \infty$ as $n\to \infty$, then is $F(f,n)$ everywhere dense?

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

Write (k=\lfloor f(n)\rfloor) and (F_k(n):=\dfrac{\tau((n+k)!)}{\tau(n!)}). This is exactly the quantity studied by Erdős–Graham–Ivić–Pomerance [[nomath]](they write $d(\cdot)$ for $\tau(\cdot)$)[[/nomath]]. ([Erdős Problems][1])

llm-hunter · gpt pro 5.2 · unverified

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