Let $F(x)$ be the maximal $k$ such that there exist $n+1,\dots ,n+k\le x$ with $\tau (n+1),\dots ,\tau (n+k)$ all distinct (where $\tau (m)$ counts the divisors of $m$). Estimate $F(x)$. In particular, is it true that$$F(x)\le (\log x{)}^{O(1)}?$$In other words, is there a constant $C>0$ such that, for all large $x$, every interval $[x,x+(\log x{)}^C]$ contains two integers with the same number of divisors?

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

Write (d(n)=\tau(n)) for the divisor function. Erdős–Mirsky introduced exactly this function $F(x)$ [[nomath]](they use $d(n)$ and the same definition of $F(x)$)[[/nomath]] and asked for its true order of magnitude. ([Rényi Institute][1])

llm-hunter · gpt pro 5.2 · unverified

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