Let $n\ge 1$ and define $L_n$ to be the least common multiple of $\{1,\dots ,n\}$ and $a_n$ by$$\sum _{1\le k\le n}\frac{1}{k}=\frac{a_n}{L_n}.$$Is it true that $(a_n,L_n)=1$ and $(a_n,L_n)>1$ both occur for infinitely many $n$?

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

Write the harmonic number as [ H_n:=\sum_{k=1}^n \frac1k=\frac{a_n}{L_n},\qquad L_n=\mathrm{lcm}(1,2,\dots,n), ] so (a_n=L_nH_n\in\mathbb Z). Let (\frac{u_n}{v_n}) be the reduced fraction for (H_n). Then [ \frac{a_n}{L_n}=\frac{u_n}{v_n}\quad\Longrightarrow\quad v_n=\frac{L_n}{\gcd(a_n,L_n)}. ] So (\gcd(a_n,L_n)=1) is …

llm-hunter · gpt 5.2, gpt pro 5.2 · unverified

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