New congruences for overcubic partition function

Abstract

In 2010, Byungchan Kim introduced a new class of partitionfunction $\overline{a}(n)$, the number of overcubic partitions of$n$ and established $\overline{a}(3n+2)\equiv 0\pmod{3}$. Our goalis to consider this function from an arithmetic point of view inthe spirit of Ramanujan's congruences for the unrestrictedpartition function $p(n)$. We prove a number of results for$\overline{a}(n)$, for example, for $\alpha \ge 0$ and $n \ge 0$,$\overline{a}(8n+5)\equiv 0\pmod{16}$, $\overline{a}(8n+7)\equiv0\pmod{32}$, $\overline{a}(8\cdot3^{2\alpha+2}n+3^{2\alpha+2})\equiv 3^{\alpha}\overline{a}(8n+1)\pmod{8}$.