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Claim. P. Erdős, On consecutive integers, Nieuw Arch. Wisk. (3) 3 (1955), 124--128, Theorem 1, printed p. 124: "There is a constant $c_1>1$ so that $f(k)\le c_1\frac{k}{\log k}$ (1). In other words the sequence $u+1,u+2,\ldots,u+t$, $t=[c_1\frac{k}{\log k}]$, $u\ge k$ has at least one prime $>k$" (result page), where $f(k)$ is the function of Problem 961. The paper leaves $c_1$ unspecified; the constant $3$ in the site's display $f(k)<3k/\log k$ is Erdős's 1976 restatement of this theorem. The journal record gives the year and no day, so this page carries the first of January.

Covers. The upper bound $f(k)\le c_1k/\log k$ for some absolute constant $c_1>1$; the order of $f(k)$ stays open.

Depends on. No page of this wiki.

Acceptance. Refereed: Nieuw Archief voor Wiskunde. The site labels the problem OPEN, so its commentary gives no reviewed evidence.