Gaps between \(r\)-free integers (Q1337791)

For \(r \geq 2\) let \(q_ r(n)\) be the \(n\)-th \(r\)-free integer (number not divisible by any \(r\)-th power other than 1). It is proved that \(q_ r (n+1) - q_ r(n) = O(n^{\alpha + \varepsilon})\) as \(n \to \infty\), with \(\alpha = 229/348(r+1)\). In particular, if \(q_ 2(n)\) is the \(n\)-th squarefree integer then \(\alpha = 229/1044 = 0.219348659 < 1057/4785\), that is \(q_ 2(n + 1) - q_ 2(n) = O(n^{229/1044 + \varepsilon})\).