In this paper, we prove some divisibility properties of sums of 24-regular partition numbers. For example, if <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_799_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$b_{24}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>b</mi> <mn>24</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denotes the number of 24-regular partitions of a positive integer n, then for non-negative integer s and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_799_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="139" /> </InlineMediaObject> <EquationSource Format="TEX">$\omega (k)=k(3k+1)/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ω</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>: <Equation ID="Equ164"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="13226_2025_799_Article_Equ164.gif" Format="GIF" Height="49" Rendition="HTML" Resolution="72" Type="Linedraw" Width="504" /> </MediaObject> <EquationSource Format="TEX">$\sum _{k=0}^{\infty }b_{24}(12s+11-\omega (-2k))+\sum _{k=1}^{\infty }b_{24}(12s+11-\omega (2k))\equiv 0\pmod {72}.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>∞</mi> </munderover> <msub> <mi>b</mi> <mn>24</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>12</mn> <mi>s</mi> <mo>+</mo> <mn>11</mn> <mo>-</mo> <mi>ω</mi> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <munderover> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </munderover> <msub> <mi>b</mi> <mn>24</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>12</mn> <mi>s</mi> <mo>+</mo> <mn>11</mn> <mo>-</mo> <mi>ω</mi> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≡</mo> <mn>0</mn> <mspace width="10.0pt" /> <mrow> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>72</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation>

Divisibility Properties of Sums of 24-Regular Partition Numbers

期刊论文
发表日期：

2025年6月19日

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Sabi Biswas,
Nipen Saikia

摘要

In this paper, we prove some divisibility properties of sums of 24-regular partition numbers. For example, if $b_{24}(n)$ $b_{24} (n)$ denotes the number of 24-regular partitions of a positive integer n, then for non-negative integer s and $\omega (k)=k(3k+1)/2$ $ω (k) = k (3 k + 1) / 2$ : $\sum _{k=0}^{\infty }b_{24}(12s+11-\omega (-2k))+\sum _{k=1}^{\infty }b_{24}(12s+11-\omega (2k))\equiv 0\pmod {72}.$ $\sum_{k = 0}^{\infty} b_{24} (12 s + 11 - ω (- 2 k)) + \sum_{k = 1}^{\infty} b_{24} (12 s + 11 - ω (2 k)) \equiv 0 (mod 72) .$