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 $.
Citation: Jinyun Qi, Zhefeng Xu. Almost primes in generalized Piatetski-Shapiro sequences[J]. AIMS Mathematics, 2022, 7(8): 14154-14162. doi: 10.3934/math.2022780
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 $.
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Jinyun Qi, Zhefeng Xu. Almost primes in generalized Piatetski-Shapiro sequences[J]. AIMS Mathematics, 2022, 7(8): 14154-14162. doi: 10.3934/math.2022780
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