Wu-Xia Ma, Yong-Gao Chen
Let $c_{k,j}(n)$ be the number of $(k,j)$-colored partitions of $n$. In 2021, Keith proved the following results: For $j=2,5,8,9$, we have $c_{9,j}(3n+2)\equiv 0\pmod {27}$ for all integers $n\ge 0$. For $j\in\{3,6\}$, we have $c_{9,j}(9n+2)\equiv 0\pmod {27}$ for all integers $n\ge 0$. Let $a,b$ be coprime positive integers. Recently, the authors gave the necessary and sufficient conditions for $c_{9,j}(an+b)\equiv 0\pmod {27}$ for all integers $n\ge 0$. In particular, for $j=1,4,7$, there does not exist coprime positive integers $a,b$ such that $c_{9,j}(an+b)\equiv 0\pmod {27}$ for all integers $n\ge 0$. In this paper, we study the congruences of $c_{4,j}(n)$. For $1\le j\le 3$, we determine all coprime positive integers $a,b$ such that $c_{4,j}(an+b)\equiv 0\pmod {8}$ for all integers $n\ge 0$.
