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Claim. For real $\alpha$ and integer $u\ge1$ let $G_\alpha(u)$ be the minimum, over all sets of $u$ distinct primes with largest member $P$ and all blocks of $\lfloor\alpha P\rfloor$ consecutive positive integers, of the number of integers in the block divisible by at least one of the primes, each integer counted once. Equivalently, $G_\alpha(u)$ is the least value of $F_{\lfloor\alpha p_u\rfloor}(p_1,\ldots,p_u)$ over all choices of primes $p_1<\cdots<p_u$. For every real $\alpha$ with $2<\alpha<3$ and every integer $u\ge1$,

$$G_\alpha(u)=\lceil2\sqrt u\rceil.$$

Covers. The estimate asked by Problem 1143 in the range $k=\alpha p_u$, for the least value over choices of the primes: that value exactly, for $2<\alpha<3$ and every $u$, extending the perfect-square case of Erdős and Selfridge. It does not compute $F_k$ for a prescribed set of primes and says nothing about $\alpha\ge3$.

Credit. The claim is a rectangular form of the construction in Theorem 1 of Section 6 of Erdős's 1978 paper, joint work with Selfridge, given in full with the prime number theorem and the Chinese remainder theorem; the exposition and the extension to every $u$ are author-recorded by Plasma AI, with no novelty claimed.

Standing. L37 is tier 0: author-recorded by Plasma AI, with a complete written proof and no independent examination or Lean proof. Catalog acceptance is pending.

Depends on. L37 (tier 0).