On 3 and 9-regular cubic partitions

Abstract

Let $a_3(n)$ and $a_9(n)$ are 3 and 9-regular cubic partitions of $n$. In this paper, we find the infinite family of congruences modulo powers of 3 for $a_3(n)$ and $a_9(n)$ such as \[a_3\left (3^{2α}n+\frac{3^{2α}-1}{4}\right )\equiv 0 \pmod{3^α}\] and \[a_9\left (3^{α+1}n+3^{α+1}-1\right )\equiv 0 \pmod{3^{α+1}}.\]