Searching for consecutive integers divisible by a power of their largest prime factor

Abstract. Given an integer \(n\ge 2\), let \(P(n)\) stand for its largest prime factor. Given integers \(k\ge 2\) and \(\ell\ge 2\), consider the set \(E_{k,\ell}\) of those integers \(n\ge 2\) for which \(P(n+i)^\ell\mid n+i\) for \(i=0,1,\ldots,k-1\). These sets are very thin. For instance, the smallest element of \(E_{3,2}\) is
1 294 298, whereas the smallest known element of \(E_{3,3}\) has 77 digits. The study of the sets \(E_{k,\ell}\) originated in 2009 and was later pursued by others. Here, we present a survey of the results obtained so far and provide new ones. In the process, given an arbitrary integer \(\ell\ge 2\) and setting \(G_\ell:=\{n\in {\mathbb N}: P(n)^\ell\mid n\}\), we introduce two polynomial-time algorithms each providing the list of all elements of \(G_\ell\) below a given bound. We conclude by raising various open problems.

Key words and phrases. Factorization, polynomials, friable numbers, Dickman's function, Mason's theorem.