A SURVEY ON k–FREENESS

Abstract. We say that an integer n is k–free (k ≥ 2) if for every prime p the valuation vp(n) < k. If f : N → Z, we consider the enumerating function S f (x) defined as the number of positive integers n ≤ x such that f(n) is k–free. When f is the identity then S f (x) counts the k–free positive integers up to x. We review the history of S f (x) in the special cases when f is the identity, the characteristic function of an arithmetic progression a polynomial, arithmetic. In each section we present the proof of the simplest case of the problem in question using exclusively elementary or standard techniques.