Dense primitive polynomial sequences

Abstract

A sequence { a n } of integers is said to be primitive if a i × a j whenever i ≠ j . Let f be a polynomial with integer coefficients and A a sequence of positive integers. We discuss further a problem considered in [1] in which I. Anderson, W. W. Stothers and the author investigated primitive sequences of the form f ( A ) = { f ( x ), x ∈ A } . (Of course, we can assume f(x)→ ∞ as x → ∞.) We shall prove the following theorem in which A(n) , as usual, denotes the number of memhers of A that are. less than or equal to n .