## Conventions ##
We will call the following product the *canonical expansion of $a$*: 
$$a = \prod\limits_{p_i\in\mathbb{P}}p_i^{\alpha_i}.$$
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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 $a$.
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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 $a$.
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Now I am trying to prove that $\beth_k$ is a multiplicative function.