Erdős Problem #860

There are distinct integers \(a_1,\ldots,a_{\pi(n)}\) in the open interval \((m,m+L)\) such that \(p_i \mid a_i\) for every \(1 \leq i \leq \pi(n)\), where \(p_i\) is the \(i\)th prime. Primes are indexed from \(0\) by `Nat.nth Nat.Prime`, so \(p_i\) is `Nat.nth Nat.Prime (i - 1)`.

\(h(n)\): the least \(L\) such that for every \(m \geq 1\) the open interval \((m,m+L)\) contains distinct integers \(a_1,\ldots,a_{\pi(n)}\) with \(p_i \mid a_i\), where \(p_i\) is the \(i\)th prime. The set of such \(L\) is nonempty: \(L = n\pi(n)+1\) works, since each \(p_i \leq n\) has at least \(\pi(n)\) multiples in \((m,m+L)\), so the \(a_i\) can be chosen one by one. So `sInf` is the true minimum. For \(n < 2\) there are no primes and \(h(n) = 0\).