Arithmetic Properties of Overpartition Triples

Liuquan Wang

Published 2014-10-29Version 1

Let ${{\bar{p}}_{3}}(n)$ be the number of overpartition triples of $n$. We find some congruences for ${\bar{p}}_{3}(n)$ modulo small powers of 2, such as \[{{\bar{p}}_{3}}(16n+14)\equiv 0 \pmod{32}, \quad {{\bar{p}}_{3}}(8n+7)\equiv 0 \pmod{64}.\] We also establish many arithmetic properties for ${{\bar{p}}_{3}}(n)$ 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 ${{\bar{p}}_{3}}\big({{3}^{2\alpha +1}}(3n+2)\big)\equiv 0$ (mod 9) and \[{{\bar{p}}_{3}}\big({{7}^{2\alpha +1}}(7n+3)\big)\equiv {{\bar{p}}_{3}}\big({{7}^{2\alpha +1}}(7n+5)\big)\equiv {{\bar{p}}_{3}}\big({{7}^{2\alpha +1}}(7n+6)\big)\equiv 0 \pmod{7}.\]