Conventions

We will call the following product the canonical expansion of $a$: $$a = \prod\limits_{p_i\in\mathbb{P}}p_i^{\alpha_i}.$$

If $a$ and $b$ are co-prime we write $$a \perp b.$$

$\beth$-function

Suppose $a\in \mathbb{N}_+$ and $M_k(a)$ is the following set: $$M_k(a) = \Big\{ (x_1,\ldots,x_k) \Big| \sum x_i = a\quad \mathrm{and}\quad x_i\perp x_j\quad \mathrm{for}\quad i \neq j\Big\}.$$ Then let us construct function (\beth): $$\beth_k(a) = |M_k(a)|.$$ We choose by definition $$\beth_k(1) = 1.$$ In other words, $\beth_k(a)$ counts co-prime $k$-partitions of $a$.

So my question is to investigate $\beth$-function's properties. The main question, no doubt, is

Write the explicit (closed form) expression for $\beth_k(a)$ using the canonical expansion of $a$.

Now I am trying to prove that $\beth_k$ is a multiplicative function.