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Claim. For $f(n)$ the least $m$ such that $n!=a_1\cdots a_k$ with $n<a_1<\cdots<a_k=m$, as in Problem 390, Samuel Mausberg's note "A Thirteen-Layer Lower Bound for Erdős Problem #390" (dated 2026-05-02, four pages, posted in the problem's discussion thread) proves as its Theorem 1 that

$$\liminf_{n\to\infty}\frac{f(n)-2n}{n/\log n}\geq C_0,\qquad C_0=\frac{\sum_{r=1}^{13}1/((r+1)(2r+1))}{\sum_{p\leq23}1/(p-1)} =\frac{4029639598}{25970038185}=0.15516\ldots$$

The proof works in the complement form: $f(n)\le M$ holds exactly when $M!/(n!)^2$ is a product of distinct integers in $(n,M]$. For $M=2n+h$ with $h=O(n/\log n)$, every prime $P$ in one of the thirteen layers $M/(2r+2)<P\le M/(2r+1)$, $n/(r+1)<P\le n/r$ ($1\le r\le13$) divides the quotient exactly once, so it sits in exactly one factor $Pq$, and the cofactor $q$ lies in $[r+1,2r+1]\subset[2,27]$ and so carries a prime at most $23$; the layers hold $(\sum_{r\le13}1/((r+1)(2r+1))+o(1))\,n/\log n$ primes by the prime number theorem, while the total valuation of the quotient at the primes up to $23$ is $h\sum_{p\le23}1/(p-1)+O(\log n)$, and comparing the two gives $h\ge(C_0-o(1))\,n/\log n$. The constant improves the lower-bound constant of Erdős, Guy and Selfridge, which could be taken arbitrarily close to $1/9$ ([EGS82], carded at Erdős, Guy and Selfridge 1982), and quantifies an obstruction Tao had described in the same thread. The digest and result list are on the library card Mausberg 2026.

Covers. The lower bound only: $\liminf(f(n)-2n)\log n/n\ge C_0$, so a constant $c$ with $f(n)-2n\sim cn/\log n$, if one exists, is at least $C_0$. The note's Remark 1 says that its bookkeeping is a necessary condition only, and the note claims no upper bound and no asymptotic formula; whether the constant exists, and its value, are the subject of the pending full claim Wang 2026, whose lower bound rests on this note.

Depends on. No page of this wiki.

Claimant and system. Mausberg posted the note in the problem's discussion thread on 2026-05-02, saying that the result was obtained with AI; the note's disclosure says it was prepared by Mausberg with the assistance of GPT-5.5 Pro over several sessions, with Mausberg responsible for the mathematical claims. The note has no arXiv record and is held on the author's Google Drive.

Standing. The site labels the problem OPEN (LEAN) and its remarks do not mention the note; a commenter in the thread reported the next day that a check had found no issues, which is not an acceptance record. The note is not refereed and has no Lean formalization of its own. The claim is claimed; it does not settle the problem, whose standing comes from the pending full claim.