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Claim. Theorem 2.1 of the note "An improved lower bound on Erdős Problem #962": for every fixed $c$ with $0<c<1/\sqrt2$ there is $n_0(c)$ such that for all $n\ge n_0(c)$, $k(n)\ge\lfloor\exp(c\sqrt{\log n\log\log n})\rfloor$. Its Theorem 1.2 is the form $k(n)\ge\exp((1/\sqrt2-o(1))\sqrt{\log n\log\log n})$. The proof is a pigeonhole lemma (if fewer than $\lfloor n/y\rfloor$ integers up to $n$ are $y$-smooth, some block of $y$ consecutive integers below $n$ has no $y$-smooth member) with the de Bruijn--Hildebrand estimate $\Psi(n,y)=n\rho(u)(1+o(1))$ and $\rho(u)=u^{-u+o(u)}$, cited from Granville's 2008 survey. The same bound follows from Erdős's 1976 display (6) through the inverse $k(n)=\max\{k:n_k\le n\}$, as Erdős 1976 records; the site credits Tang separately with the explicit constant.

Covers. The lower bound on $k(n)$ only; neither question of Problem 962 is settled by it.

Claimant and postings. Quanyu Tang posted the note, a PDF in the repository QuanyuTang/erdos-problem-962, to the site's thread on 28 December 2025; the PDF's metadata dates it 28 December 2025 (UTC). The formal-conjectures file for the problem states the bound as the variant tang_lower_bound, with a sorry body.

Depends on. No page of this wiki.

Standing. Claimed. The note has no refereed version, and the site's commentary on a problem it labels OPEN is not acceptance.