Abstract
Let S be a finite set of rational primes, and let sn denote the increasing sequence of the positive integers having all their prime factors in S. In this paper we develop a method to explicitly give the gaps in the sequence sn. In other words, for any term sn we can find both sn−1 and sn+1, at least in principle, without enumerating all terms of the sequence. In the case when S contains two fixed primes, we even give an efficient algorithm to find these terms explicitly. Further, we apply our results to prove some Diophantine properties of the sequence sn.
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