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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 Sk f (x) defined as the number of positive integers n ≤ x such that f(n) is k–free. When f is the identity then Sk f (x) counts the k–free positive int ..."
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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 Sk f (x) defined as the number of positive integers n ≤ x such that f(n) is k–free. When f is the identity then Sk f (x) counts the k–free positive integers up to x. We review the history of Sk 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. 1. Introduction The
Sieve Methods
"... Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after th ..."
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Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum