  Suppose that $\alpha,\beta\in\mathbb{R}$. Let $\alpha\geqslant1$ 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.
