Suppose that \(\alpha ,\beta \in \mathbb {R}\) . Let \(\alpha \geqslant 1\) and c be a real number in the range \(1<c< 12/11\) . In this paper, it is proved that there exist infinitely many primes in the generalized Piatetski–Shapiro sequence, which is defined by \((\lfloor \alpha n^c+\beta \rfloor )_{n=1}^\infty \) . Moreover, we also prove that there exist infinitely many Carmichael numbers composed entirely of primes from the generalized Piatetski–Shapiro sequences with \(c\in (1,\frac{19137}{18746})\) . The two theorems constitute improvements upon previous results by Guo and Qi [5].
