Product of an integer free of small prime factors and prime in arithmetic progression

Abstract: We establish estimates for the number of ways to represent any reduced residue class as a product of a prime and an integer free of small prime factors. Our best results is conditional on the Generalised Riemann hypothesis (GRH). As a corollary, we make progress on a conjecture of Erd\"os, Odlyzko and S\'ark\"ozy.