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 $α\ge 1$ and $n \ge 0$, we have ${\overline{p}_{3}}\big({{3}^{2α+1}}(3n+2)\big)\equiv 0$ (mod $9\cdot 2^4$), $\overline{p}_{3}(4^{α-1}(56n+49)) \equiv 0$ (mod 7) and \[{\overline{p}_{3}}\big({{7}^{2α+1}}(7n+3)\big)\equiv {\overline{p}_{3}}\big({{7}^{2α+1}}(7n+5)\big)\equiv {\overline{p}_{3}}\big({{7}^{2α+1}}(7n+6)\big)\equiv 0 \pmod{7}.\]