Abstract
We consider a generalization of Piatetski–Shapiro sequences in the sense of Beatty sequences, which is of the form $(\lfloor \alpha n^c + \beta \rfloor)_{n=1}^{\infty}$ with real numbers $\alpha \geq 1$, $c \gt 1$ and $\beta$. We show there are infinitely many primes in the generalized Piatetski–Shapiro sequence with $c \in (1,14/13)$. Moreover, we prove there are infinitely many Carmichael numbers composed entirely of the primes from the generalized Piatetski–Shapiro sequences with $c \in (1,64/63)$.
Funding Statement
The first author is supported in part by the National Natural
Science Foundation of China (No. 11901447), the China Postdoctoral Science Foundation
(No. 2019M653576) and the Natural Science Foundation of Shaanxi Province (No. 2020JQ-009). The
second author is supported in part by the National Natural Science Foundation of China
(No. 11971381, No. 11701447, No. 11871317 and No. 11971382).
Acknowledgments
The authors thank the referees for their valuable comments. The authors also thank Prof. Yuan Yi and Prof. Yaming Lu for several helpful discussions.
Citation
Victor Zhenyu Guo. Jinyun Qi. "A Generalization of Piatetski–Shapiro Sequences." Taiwanese J. Math. 26 (1) 33 - 47, February, 2022. https://doi.org/10.11650/tjm/210802
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