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<p>Utilizing high-performance computing, lookup tables, and the property of coalescence with respect to Collatz sequences, we present an approach to computing streaks of consecutive integers with the same Collatz height. Of particular interest is the extension of the OEIS (On-Line Encyclopedia of Integer Sequences) sequence A277109 through element 3950. At the time of publication, 9336 of the first 10000 elements of this sequence have been established. The current longest confirmed streak is for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11227_2025_7820_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">\(2^{3951}+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mn>3951</mn> </msup> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, with a streak of length <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11227_2025_7820_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="122" /> </InlineMediaObject> <EquationSource Format="TEX">\(2^{47}-1 \approx 10^{14.148}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mn>2</mn> <mn>47</mn> </msup> <mo>-</mo> <mn>1</mn> <mo>≈</mo> <msup> <mn>10</mn> <mrow> <mn>14.148</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Computing streaks of consecutive numbers with the same Collatz height

  • 期刊论文

  • 发表日期：

    2025年9月15日

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  • Karl Frinkle,
  • Saxon Rowland,
  • James Pack,
  • Madison Gordon,
  • John Eskue,
  • Sardonyx Egbe,
  • Mike Morris

摘要

Utilizing high-performance computing, lookup tables, and the property of coalescence with respect to Collatz sequences, we present an approach to computing streaks of consecutive integers with the same Collatz height. Of particular interest is the extension of the OEIS (On-Line Encyclopedia of Integer Sequences) sequence A277109 through element 3950. At the time of publication, 9336 of the first 10000 elements of this sequence have been established. The current longest confirmed streak is for \(2^{3951}+1\) 2 3951 + 1 , with a streak of length \(2^{47}-1 \approx 10^{14.148}\) 2 47 - 1 ≈ 10 14.148 .