3-Divisibility of 9- and 27-Regular Partitions

A partition of $n$ is $l$-regular if none of its parts is divisible by $l$. Let $b_l(n)$ denote the number of $l$-regular partitions of $n$. In this paper, using the theory of Hecke eigenforms explored by J.-P. Serre, we establish exact criteria for the $3$-divisibility of $b_9(n)$ and $b_{27}(n)$.