Arithmetic Properties of Overpartition Triples

Abstract

Let ${\overline{p}}_3(n)$ be the number of overpartition triples of $n$. By elementary series manipulations, we establish some congruences for ${\overline{p}}_3(n)$ modulo small powers of 2, such as \[{\overline{p}_{3}}(16n+14)\equiv 0 \pmod{32}, \quad {\overline{p}_{3}}(8n+7)\equiv 0 \pmod{64}.\] We also find many arithmetic properties for ${\overline{p}}_3(n)$ modulo 7, 9 and 11, involving the following infinite families of Ramanujan-type congruences: for any integers $\alpha \ge 1$ and $n\ge 0$, we have ${\overline{p}}_3(3^{2\alpha +1}(3n+2))\equiv 0$ (mod $9\cdot 2^4$), ${\overline{p}}_3(4^{\alpha -1}(56n+49))\equiv 0$ (mod 7) and \[{\overline{p}_{3}}\big({{7}^{2α+1}}(7n+3)\big)\equiv {\overline{p}_{3}}\big({{7}^{2α+1}}(7n+5)\big)\equiv {\overline{p}_{3}}\big({{7}^{2α+1}}(7n+6)\big)\equiv 0 \pmod{7}.\]

14 pages. We corrected some typos in the first version. Some new results have been added