Harmonic-sum numerator vs. $\mathrm{lcm}(1,\ldots,n)$: do coprime and non-coprime cases each occur infinitely often? (Erdős #291)

Let $L_n=\mathrm{lcm}(1,2,\ldots,n)$ and write the $n$-th harmonic number as $\sum_{1\le k\le n}\frac1k=\frac{a_n}{L_n}$ with $a_n=\sum_{k\le n}L_n/k\in\mathbb{Z}$ (not necessarily reduced). Is it true that both $\gcd(a_n,L_n)=1$ and $\gcd(a_n,L_n)>1$ occur for infinitely many $n$?

Acceptance. PARTIAL / EXTENDS: compute $g_n=\gcd(a_n,L_n)$ for $n$ up to a large bound, tabulate the $n$ with $g_n>1$ and with $g_n=1$, and report observed densities (evidence each set is infinite). FULLY RESOLVES: a proof that each of $\{n:\gcd(a_n,L_n)=1\}$ and $\{n:\gcd(a_n,L_n)>1\}$ is infinite. Provide the computation.