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Smooth numbers in Beatty sequences

Volume 200 / 2021

Roger Baker Acta Arithmetica 200 (2021), 429-438 MSC: Primary 11N25; Secondary 11L03. DOI: 10.4064/aa210322-22-6 Published online: 6 October 2021

Abstract

Let $\theta $ be an irrational number of finite type and let $\psi \ge 0$. We consider numbers in the Beatty sequence of integer parts, \[ \mathcal B(x) = \{\lfloor \theta n + \psi \rfloor : 1 \le n \le x\}. \] Let $C \gt 3$. Writing $P(n)$ for the largest prime factor of $n$ and $|\ldots |$ for cardinality, we show that \[ |\{n\in \mathcal B(x) : P(n) \le y\}| = \frac 1\theta \, \Psi (\theta x, y) (1 + o(1)) \] as $x\to \infty $, uniformly for $y \ge (\log x)^C$. Here $\Psi (X,y)$ denotes the number of integers up to $X$ with $P(n) \le y$. The range of $y$ extends that given by Akbal (2020). The work of Harper (2016) plays a key role in the proof.

Authors

  • Roger BakerDepartment of Mathematics
    Brigham Young University
    Provo, UT 84602, U.S.A.
    e-mail

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