Abstract
Suppose that \(\alpha ,\beta \in \mathbb {R}\). Let \(\alpha \geqslant 1\) 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 [5].
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Acknowledgements
The authors would like to appreciate the referee for his/her patience in refereeing this paper. This work is supported by the National Natural Science Foundation of China (Grants No. 12001047, 11901566, 11971476, 12071238, 11971381, 11701447, 11871317, 11971382), and the Natural Science Foundation of Beijing Municipal (Grant No. 1242003).
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Communicated by B. Sury.
Jinyun Qi is the corresponding author.
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Li, J., Qi, J. & Zhang, M. A generalization of Piatetski–Shapiro sequences (II). Indian J Pure Appl Math (2024). https://doi.org/10.1007/s13226-024-00532-4
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DOI: https://doi.org/10.1007/s13226-024-00532-4