The sequence $$ \mathbb{ P}^{ (c)}=(\lfloor p^c \rfloor)_{ p\in \mathbb{ P}}\quad (c>0,c\notin \mathbb{ N}), $$ is an important subsequence of the well-known Piatetski-Shapiro sequence, where $\mathbb{ P}$ is the set of prime numbers and $\lfloor \cdot \rfloor$ is the floor function. We prove that for all $c \in (0, 13/15)$, any large enough integer $N$ can be represented as $$ N=\lfloor p^c\rfloor+q, $$ where $p$ and $q$ are primes. We also prove the result holds for almost all fixed positive $c \in \mathbb{ R}\setminus\mathbb{ Z}$. Moreover, we investigate shifted primes in this sequence, obtaining an asymptotic formula for all $c \in (0, 13/15)$ and an almost-all result for fixed positive $c \in \mathbb{ R}\setminus\mathbb{ Z}$.
Categories: math.NT