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<p>Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal M\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">M</mi> </math></EquationSource> </InlineEquation> be a set of positive integers with the minimal element <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(m\ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. A positive integer is called an <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\mathcal M\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">M</mi> </math></EquationSource> </InlineEquation><i>-free integer</i> if every exponent of its prime factors is not contained in <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathcal M\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">M</mi> </math></EquationSource> </InlineEquation>. We study the distribution of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\mathcal M\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">M</mi> </math></EquationSource> </InlineEquation>-free integers in the Piatetski-Shapiro sequences. Several new results of the distribution of special integers in the Piatetski-Shapiro sequences also are obtained.</p>

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On the distribution of \(\mathcal M\)-free integers in Piatetski-Shapiro sequences

  • 期刊论文

  • 发表日期：

    2025年12月4日

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  • Nithi Rungtanapirom,
  • Saeree Wananiyakul,
  • Teerapat Srichan

摘要

Let \(\mathcal M\) M be a set of positive integers with the minimal element \(m\ge 2\) m ≥ 2 . A positive integer is called an \(\mathcal M\) M -free integer if every exponent of its prime factors is not contained in \(\mathcal M\) M . We study the distribution of \(\mathcal M\) M -free integers in the Piatetski-Shapiro sequences. Several new results of the distribution of special integers in the Piatetski-Shapiro sequences also are obtained.