In 2024, Nayaka, Dharmendra and Mahesh Kumar introduced the restricted overcubic partition triple function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\overline{bt}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mrow> <mi mathvariant="italic">bt</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which counts the overlined versions of cubic partition triples of a positive integer n. In this paper, we obtain several infinite families of congruences modulo 4 and 8 for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\overline{bt}(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mrow> <mi mathvariant="italic">bt</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> by employing standard tools from the theory of modular forms, including eta-products and Hecke operators. For example, <Equation ID="Equ85"> <EquationSource Format="TEX">$\begin{aligned} \overline{bt}(32\cdot 3^{4\alpha +2}n+68\cdot 3^{4\alpha +1})\equiv 0\pmod {8} \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mover> <mrow> <mi mathvariant="italic">bt</mi> </mrow> <mo>¯</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mn>32</mn> <mo>·</mo> <msup> <mn>3</mn> <mrow> <mn>4</mn> <mi>α</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mi>n</mi> <mo>+</mo> <mn>68</mn> <mo>·</mo> <msup> <mn>3</mn> <mrow> <mn>4</mn> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <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>8</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for all nonnegative integers <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> and n.

New modular congruences for overcubic partition triples

期刊论文
发表日期：

2026年6月19日

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S. Shivaprasada Nayaka

摘要

In 2024, Nayaka, Dharmendra and Mahesh Kumar introduced the restricted overcubic partition triple function $\overline{bt}(n)$ $\bar{bt} (n)$ , which counts the overlined versions of cubic partition triples of a positive integer n. In this paper, we obtain several infinite families of congruences modulo 4 and 8 for $\overline{bt}(n)$ $\bar{bt} (n)$ by employing standard tools from the theory of modular forms, including eta-products and Hecke operators. For example, $\begin{aligned} \overline{bt}(32\cdot 3^{4\alpha +2}n+68\cdot 3^{4\alpha +1})\equiv 0\pmod {8} \end{aligned}$ $\begin{matrix} \bar{bt} (32 \cdot 3^{4 α + 2} n + 68 \cdot 3^{4 α + 1}) \equiv 0 (mod 8) \end{matrix}$ for all nonnegative integers $\alpha $ $α$ and n.