Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> be an irrational number, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> and c be real numbers. The Piatetski–Shapiro sequence and the corresponding non-homogeneous Beatty sequence are defined as <Equation ID="Equ24"> <EquationSource Format="TEX">$ \mathcal {N}^{(c)}=(\left\lfloor n^c\right\rfloor )_{n=1}^{\infty } \quad (c>1, c \notin \mathbb {N}),\quad \text {and}\quad \mathcal {B}_{\alpha , \beta }=(\left\lfloor \alpha n+\beta \right\rfloor )_{n=1}^{\infty } $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msup> <mrow> <mi mathvariant="script">N</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msubsup> <mrow> <mo stretchy="false">(</mo> <mfenced close="⌋" open="⌊"> <msup> <mi>n</mi> <mi>c</mi> </msup> </mfenced> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> <mspace width="1em" /> <mrow> <mo stretchy="false">(</mo> <mi>c</mi> <mo>></mo> <mn>1</mn> <mo>,</mo> <mi>c</mi> <mo>∉</mo> <mi mathvariant="double-struck">N</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mtext>and</mtext> <mspace width="1em" /> <msub> <mi mathvariant="script">B</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mrow> <mo stretchy="false">(</mo> <mfenced close="⌋" open="⌊"> <mi>α</mi> <mi>n</mi> <mo>+</mo> <mi>β</mi> </mfenced> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </mrow> </math></EquationSource> </Equation>respectively. For every <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$R > 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, we say that a natural number is an R-almost prime if it has at most R prime factors, counted with multiplicity. In this paper, we prove that there are infinitely many Beatty primes of the form <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\lfloor n^c \rfloor $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>⌊</mo> <msup> <mi>n</mi> <mi>c</mi> </msup> <mo>⌋</mo> </mrow> </math></EquationSource> </InlineEquation> such that n is an R-almost primes with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$c \in (1, c_{R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>c</mi> <mi>R</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$c_{R}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>c</mi> <mi>R</mi> </msub> </math></EquationSource> </InlineEquation> is an explicit constant depending on R.

Special primes from almost-primes

期刊论文
发表日期：

2026年7月18日

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Lingyu Guo,
Victor Zhenyu Guo,
Kunyue Li,
Xiaoying Liu,
Li Lu

摘要

Let $\alpha $ $α$ be an irrational number, $\beta $ $β$ and c be real numbers. The Piatetski–Shapiro sequence and the corresponding non-homogeneous Beatty sequence are defined as $ \mathcal {N}^{(c)}=(\left\lfloor n^c\right\rfloor )_{n=1}^{\infty } \quad (c>1, c \notin \mathbb {N}),\quad \text {and}\quad \mathcal {B}_{\alpha , \beta }=(\left\lfloor \alpha n+\beta \right\rfloor )_{n=1}^{\infty } $ $N^{(c)} = {(⌊n^{c}⌋)}_{n = 1}^{\infty} (c > 1, c \notin N), and B_{α, β} = {(⌊α n + β⌋)}_{n = 1}^{\infty}$ respectively. For every $R > 1$ $R > 1$ , we say that a natural number is an R-almost prime if it has at most R prime factors, counted with multiplicity. In this paper, we prove that there are infinitely many Beatty primes of the form $\lfloor n^c \rfloor $ $⌊ n^{c} ⌋$ such that n is an R-almost primes with $c \in (1, c_{R})$ $c \in (1, c_{R})$ and $c_{R}$ $c_{R}$ is an explicit constant depending on R.