Parity of the 8-regular partition function

Abstract

We give a complete characterization of the parity of $b_8(n)$, the number of $8$-regular partitions of $n$. Namely, we prove that $b_8(n)$ is odd precisely when $24n+7$ has the form $p^{4a+1}m^2$ with $p$ prime and $p\nmid m$.