On the Divisibility Properties of Sequences of Integers

Abstract

Let a, < n2 < . . . be a sequence of integers denoted by A. Put A(x) = xCniCs 1. If no ai divides any other then A is called a p&m&e sequence. It is well known and easy to see that, for a primitive sequence, max A(z) = [3(x+ l)]. B esicovitch (1) constructed a primitive sequence of positive upper density and Behrend and Erd6s (1) proved that every primitive sequence has lower density 0. Davenport and Erdijs (1) proved that if A has positive upper logarithmic density then there is an infinite subsequence (ni,)j,r,. of A such that CZ,~ j aij+l. Erdijs (2) proved that’, if we assume that no ai divides the product of two others, then$