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Claim. The answer to Problem 1217 is yes. Theorem 0.1 of the nine-page manuscript A divisibility subsequence with large logarithmic counting function, posted in a public repository on 16 April 2026 (the first link above, pinned to its upload commit), states the following. Let $A=\{a_1<a_2<\cdots\}\subset\{2,3,\dots\}$ be infinite with positive lower logarithmic density. Then there is an infinite subsequence $a_{n_1}<a_{n_2}<\cdots$ with $a_{n_i}\mid a_{n_{i+1}}$ for every $i\ge1$ and

$$\limsup_{x\to\infty}\frac{\#\{i:a_{n_i}<x\}}{\log\log x}\ge \limsup_{x\to\infty}\frac{1}{\log\log x}\sum_{a_n<x}\frac{1}{a_n\log a_n}.$$

Remark 0.4 of the manuscript adds that the argument uses only the positivity of the right-hand side, so the same proof gives the conclusion under that weaker hypothesis. The proof builds a random divisibility chain from independent exponential random variables attached to the primes, bounds the probability that a given integer is visited, and extracts one realization from first- and second-moment bounds.

Claimants and assistance. The manuscript prints no author names. Quanyu Tang posted it in the site's discussion thread on 16 April 2026, saying that Tang and Yanyang Li used GPT 5.4 Pro to solve the problem, that Tang checked the proofs by hand and made only minor changes for rigor, and that Tang welcomed checks. The same day Jared Duker Lichtman reported in the thread an independent proof Lichtman was writing up.

Relation to the accepted result. This is a separate work from the eight-author preprint of 1 May 2026, of which Tang and Li are two of the authors: its claim page records Theorem 1.6 of that paper, which proves the inequality under the weaker hypothesis, and the site's page says that similar proofs were found independently by subsets of its authors and by GPT 5.4 Pro.

Depends on. No page of this wiki.

Standing. Claimed. The site's curator credits the eight-author paper, not this manuscript; the manuscript has no journal record and no Lean proof, and nothing in this repository has verified the proof.