Let \(\lfloor z\rfloor \) be the integer part of a real number z. For non-integer \(c>1\) , asymptotic formulas are derived for sums of the forms \(\begin{aligned} \sum _{n\le N}\mu ^2(n)\mu ^2(\lfloor n^c \rfloor ), \,\,\,\sum _{n\le N}\mu ^2(n)q_2(\lfloor n^c \rfloor ) \end{aligned}\) where \(\mu ^2(n)\) and \(q_2(n)\) are the characteristic functions of square-free and square-full integers, respectively. Using the exponent pair approach, it is proved that the sequence \((\lfloor n^c \rfloor )_{n\in Q_2}\) contains infinitely many square-free integers, where \(Q_2\) is the set of all square-free numbers and c belongs to a range depending on two exponent pairs. By the same method, it is shown that the sequence \((\lfloor n^c \rfloor )_{n\in Q_2}\) contains infinitely many square-full integers, where c belongs to a range depending on another two exponent pairs.
