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Claim. There is a good sequence $u_1<u_2<\cdots$ of primes, in the sense of Problem 1101, with

$$\log u_n=\sqrt n+O(n^{-1/2}),$$

so that $u_n=\exp((1+o(1))\sqrt n)\le e^{o(n)}$. This is Theorem 1 of Peter Li, A Subexponential Good Sequence, a seven-page manuscript dated 27 April 2026 (the preprint link), whose AI disclosure states that it was produced with use of GPT-5.5 Pro. It was posted to the discussion thread the same day by the user lipet2k, as a partial result obtained with help from GPT-5.5 Pro (the discussion link). The construction fixes finitely many small primes and then, for each large $n$, chooses $u_n$ independently and uniformly among the primes in $[e^{\sqrt n},e^{\sqrt n+1/(10\sqrt n)}]$; the prime number theorem with the de la Vallée Poussin error term makes these intervals disjoint and rich in primes, and a lemma on excluding certificates, proved by bounding expected counts and applying the Borel--Cantelli lemma, shows that with probability one the sequence is good.

Covers. The second question: some good sequence satisfies $u_n\le e^{o(n)}$. The first question, whether some good sequence satisfies $u_n<n^{O(1)}$, is not touched.

Depends on. No page of this wiki.

Standing. Claimed. A thread reply of 28 April 2026 reports that a check run with ChatGPT found no issues and calls the result a positive answer to the second question; that is neither a review nor acceptance. The site labels the problem OPEN (page last edited 19 October 2025), and its proof-claims tab carries no claim. No arXiv record, refereed version or outside review is known, so the claim lists no evidence.