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.(n) denote the number of representations of a non-negative integer n in the form x(1)(2) + x(2)(2) + x(3)(2) + 3y(1)(2) + 3y(2)(2) + 3y(3)(2) with x1, x2, x3, y1, y2, y3 is an element of Z. We find three arithmetic relations between B-3(n) and omega(n), such as omega(6n + 5) = 4B(3)(6n + 4).