kth powers in Piatetski-Shapiro sequences

The Piatetski-Shapiro sequences are sequences of the form [Formula: see text] with [Formula: see text] and [Formula: see text]. We investigate the distribution of [Formula: see text]th powers in Piatetski-Shapiro sequences. Moreover, we study the solution of a more general equation [Formula: see text] with [Formula: see text] for an integer [Formula: see text]. We obtain bounds for different values of [Formula: see text] and an average bound over [Formula: see text]-free numbers [Formula: see text] in an interval. In the end, we also study the solution of the equation [Formula: see text] for which [Formula: see text] is a prime number.