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Claim. Write $H_n=\sum_{k\le n}1/k=c_n/d_n$ in lowest terms and $D_n=\mathrm{lcm}(1,\ldots,n)=d_nq_n$; in the notation of Problem 291, $q_n=(a_n,L_n)$. Shiu's [../library/unit_fractions/shiu_2016_denominators_harmonic_numbers_revised/theorem_1|Theorem 1] states that $d_n<d_{n-1}$ for infinitely many $n$; since $d_{n-1}\le D_{n-1}\le D_n$, every such $n$ has $q_n>1$. Shiu's Theorem 2 is the exact criterion behind this: for an odd prime $p\le n$, $p\mid q_n$ if and only if the leading digit $m$ of $n$ in base $p$ satisfies $p\mid c_m$. Pairing $1/j$ with $1/(p-j)$ shows $p\mid c_{p-1}$ for every odd prime, so every $n\ge p$ whose leading digit in base $p$ is $p-1$ has $p\mid q_n$; for $p=3$ these are the $n$ with $2\cdot3^a\le n<3^{a+1}$, $a\ge1$, a set of positive lower density. Hence $(a_n,L_n)>1$ for infinitely many $n$: the second question of the problem is answered in the affirmative.

Covers. The second question, that $(a_n,L_n)>1$ occurs for infinitely many $n$, together with the criterion that says for which $n$ a given odd prime divides $(a_n,L_n)$. Not covered: the first question, whether $(a_n,L_n)=1$ occurs for infinitely many $n$, which the same paper states as a conjecture (with the count of such $n\le x$ of order $x/\log x$) and does not prove.

Standing. Claimed. The site's curator, Thomas Bloom, records in the problem's commentary that the second question is answered affirmatively and states the leading-digit criterion as a necessary and sufficient condition; Bloom credits the base-$3$ case to Stefan Steinerberger and cites Shiu's preprint only for the heuristic count of the complementary set, so the curator's credit goes to Steinerberger's observation, which has its own pending page, Steinerberger's base-3 observation, and is not reviewed evidence for this claim (the site labels the problem OPEN); this preprint is the earlier written proof of the same answer, disclosed on that page, and the attribution to Shiu rests on the paper itself. The paper is an unrefereed preprint (arXiv:1607.02863, v1 of 11 July 2016, v2 of 30 July 2024; no journal record was found on 2026-09-18), so refereed is not listed, and no Lean built by this corpus checks the statement, so formalized is not listed; the formal-conjectures statement file marks the corresponding part solved with a sorry body, which is not a formalization. The library's source card cites the preprint (arXiv:1607.02863v2), though no file of it is held; its result pages record the statements read clause by clause and the elementary proofs read through, and the problem page records an exact-arithmetic check of the criterion for all odd primes below $60$ and all $n\le3000$ and of the values $q_n$ against OEIS A110566 up to $n=10000$.