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Claim. Let $C(r)$ be "the maximal length $C(r)$ of a sequence of consecutive integers each divisible by one of $r$ arbitrarily chosen primes" (p. 225). The Corollary of the paper (p. 226) reads: "We have $C(r)\ll r^2\log^2r$." It follows the paper's Theorem: for an absolute $c>0$ and arbitrary primes $q_1,\ldots,q_r$, $r>1$, each interval of length $c\prod_{i\le r}(1-1/q_i)^{-1}r^2\log r$ contains at least $r^2$ integers coprime to $q_1\cdots q_r$, proved by a shifted linear sieve. For Problem 970, a run of consecutive integers each sharing a factor with $n$ is a run each divisible by one of the $\omega(n)$ primes of $n$, and $C$ is nondecreasing, so $h(k)=C(k)+1$, the one-line step recorded on the library's Corollary page; the Corollary is therefore $h(k)\ll(k\log k)^2$.

Covers. The upper bound $h(k)\ll(k\log k)^2$ only. Neither the order of magnitude of $h(k)$ nor Jacobsthal's $h(k)\ll k^2$ is settled by it.

Depends on. No page of this wiki.

Acceptance. Refereed: H. Iwaniec, On the problem of Jacobsthal, Demonstratio Math. 11 (1978), no. 1, 225--231. The site labels the problem OPEN, so its commentary crediting the bound is not reviewed evidence. The record gives the year without a day, so the day in the page name is a placeholder.