Divisibility Properties of Odd Part and Even Part Sums in Cubic Partitions

A cubic partition of a positive integer $n$ is a partition of $n$ in which even parts may appear in two distinct colors. The purpose of this paper is to study the arithmetic functions $COS(n)$ and $CES(n)$, which denote the sums of all odd and even parts, respectively, in all cubic partitions of $n$. We obtain their generating functions and divisor-sum formulas, and establish Ramanujan-type congruences modulo $3$, $4$, $5$, and $7$. To prove these congruences, we use Lambert series, classical theta-function identities, dissections, coefficient extraction, and weighted Euler identities.