New congruences for overcubic partition function

Abstract

In 2010, Byungchan Kim introduced a new class of partitionfunction $\bar{a}(n)$, the number of overcubic partitions of$n$ and established $\bar{a}(3n+2)\equiv 0(\mathrm{mod}3)$. 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$\bar{a}(n)$, for example, for $\alpha \ge 0$ and $n\ge 0$,$\bar{a}(8n+5)\equiv 0(\mathrm{mod}16)$, $\bar{a}(8n+7)\equiv 0(\mathrm{mod}32)$, $\bar{a}(8\cdot 3^{2\alpha +2}n+3^{2\alpha +2})\equiv 3^{\alpha }\bar{a}(8n+1)(\mathrm{mod}8)$.