kth powers in Piatetski-Shapiro sequences.

The Piatetski-Shapiro sequences are sequences of the form ( n c ) with c ≥ 1 and c ∉ ℕ. We investigate the distribution of k th powers in Piatetski-Shapiro sequences. Moreover, we study the solution of a more general equation n c = s m k with m , n ∈ ℕ , 1 ≤ n ≤ N for an integer s. We obtain bounds for different values of k and an average bound over k -free numbers s in an interval. In the end, we also study the solution of the equation n c = s p 2 for which p is a prime number. [ABSTRACT FROM AUTHOR]