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Claim. In the notation of Problem 291, if the leading digit of $n$ in base $3$ is $2$, that is $2\cdot3^a\le n<3^{a+1}$ for some $a\ge1$, then $3\mid(a_n,L_n)$. The reason is short: $3^a$ is the exact power of $3$ dividing $L_n$, so in $a_n=\sum_{k\le n}L_n/k$ every term with $3^a\nmid k$ is a multiple of $3$, and the only $k\le n$ divisible by $3^a$ are $3^a$ and $2\cdot3^a$, whose two terms sum to $3M$ with $M=L_n/(2\cdot3^a)$; hence $3\mid a_n$, and $3\mid L_n$ since $n\ge3$. These $n$ form a set of positive lower density, so $(a_n,L_n)>1$ 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 $n$. Not covered: the first question, whether $(a_n,L_n)=1$ occurs for infinitely many $n$, and the exact criterion saying which primes divide $(a_n,L_n)$, which the site's commentary states in general and which is Shiu's Theorem 2 on Shiu's claim page.

Standing. Claimed. The site's curator, Thomas Bloom, records in the problem's commentary that the second question has an easy affirmative answer and credits the observation to Stefan Steinerberger; but the site labels the whole problem OPEN and declares no parts, so that credit is context for this claim and not acceptance evidence, and reviewed is not listed. The observation has no written source of its own, so refereed is not listed. The site's page shows a last edit of 12 January 2026; the observation is absent from the archived copy of the page of 19 June 2024, which carries the statement and no commentary, and present in the archived copy of 7 November 2024, the date this page carries. An earlier written proof of the same answer is Shiu's preprint of 2016, whose Theorem 2 gives the general criterion and whose Theorem 1(iii) gives the infinitude by another route; the commentary cites that preprint only for a heuristic count, so it has its own pending page, linked above. A Lean 4 package submitted on 16 September 2026 as a pull request to a prize program's repository (TheJustinSunPrize/awards, PR 216, closed without merge on 24 September 2026; linked above at its head commit) proves erdos_291_part_ii, the statement of the formal-conjectures rendering erdos_291.parts.ii, from this observation (its lemma three_dvd_a), names the observation as recorded on the site and claims no novelty; its README says the proof was prepared with Claude Code, with Claude Fable 5.1 for orchestration and Claude Opus 5 for implementation. It is third-party Lean that this corpus has not built or audited, so formalized is not listed, and the formal-conjectures statement file itself marks part (ii) solved with a sorry body, which is not a formalization. The observation is the case $p=3$, $m=2$ of the criterion checked by exact arithmetic on the problem page for all odd primes below $60$ and all $n\le3000$.