Arithmetic Identities and Congruences for Partition Triples with 3-cores

Let $B_3(n)$ denote the number of partition triples of $n$ where each partition is 3-core. With the help of generating function manipulations, we find several infinite families of arithmetic identities and congruences for $B_3(n)$. Moreover, let $\omega (n)$ denote the number of representations of a nonnegative integer $n$ in the form $x_1^2+x_2^2+x_3^2+3y_1^2+3y_2^2+3y_3^2$ with $x_1,x_2,x_3,y_1,y_2,y_3\in \mathbb{Z}\mathrm{.}$ We find three arithmetic relations between $B_3(n)$ and $\omega (n)$, such as $\omega (6n+5)=4B_3(6n+4)\mathrm{.}$