Almost primes in generalized Piatetski-Shapiro sequences

Abstract

We consider a generalization of Piatetski-Shapiro sequences in the sense of Beatty sequences, which is of the form $\left( \left\lfloor{\alpha n^c + \beta}\right\rfloor \right)_{n = 1}^{\infty} $ with real numbers $ \alpha {\geqslant} 1, c > 1 $ and $ \beta $. In this paper, we prove that there are infinitely many $ R $-almost primes in sequences $ \left(\lfloor \alpha n^c + \beta \rfloor\right)_{n = 1}^{\infty} $ if $ c \in (1, c_R) $ and $ c_R $ is an explicit constant depending on $ R $.

Keywords

• piatetski-shapiro sequences
• almost primes
• exponent pair