<p>In this paper, we prove several new infinite families of Ramanujan–like congruences satisfied by the coefficients of the generating function <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1097_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$U_t(a, q)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>U</mi> <mi>t</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> which is an extension of MacMahon’s generalized sum-of-divisors function. As a by-product, we also show that, for all <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1097_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1097_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="182" /> </InlineMediaObject> <EquationSource Format="TEX">$\overline{B}_3(15n+7)\equiv 0 \pmod {5}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mi>B</mi> <mo>¯</mo> </mover> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mn>15</mn> <mi>n</mi> <mo>+</mo> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> <mo>≡</mo> <mn>0</mn> <mspace width="4.44443pt" /> <mrow> <mo stretchy="false">(</mo> <mo>mod</mo> <mspace width="0.277778em" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1097_Article_IEq4.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$\overline{B}_3(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mover> <mi>B</mi> <mo>¯</mo> </mover> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the number of almost 3-regular overpartitions of <i>n</i>.</p>

Arithmetic properties of MacMahon-type sums of divisors

期刊论文
发表日期：

2025年4月15日

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James A. Sellers,
Roberto Tauraso

摘要

In this paper, we prove several new infinite families of Ramanujan–like congruences satisfied by the coefficients of the generating function $U_t(a, q)$ $U_{t} (a, q)$ which is an extension of MacMahon’s generalized sum-of-divisors function. As a by-product, we also show that, for all $n\ge 0$ $n \geq 0$ , $\overline{B}_3(15n+7)\equiv 0 \pmod {5}$ ${\bar{B}}_{3} (15 n + 7) \equiv 0 (mod 5)$ where $\overline{B}_3(n)$ ${\bar{B}}_{3} (n)$ is the number of almost 3-regular overpartitions of n.