Primitive sequences whose elements have no large prime factors

Abstract

A sequence a1 < a2 < … of positive integers is said to be primitive if no element of the sequence divides any other. The study of primitive sequences arose naturally out of investigations into the subject of abundant numbers, where sequences each of whose elements is of the form , the pi being fixed primes, are of particular importance. Such a sequence is said to be built up from the primes p1…pr. Thus Dickson [1], in an early paper on abundant numbers, proved that a primitive sequence built up from a fixed set of primes is necessarily finite.