Apologies for the vague title, but i don't know how to give this question a proper one !
Consider the power series : $$k_{i}(y)=\sum_{m=1}^{\infty}c_{i,m}y^{im}\;\;\;\;\;\;y\in\text{ some open disk}$$ where the numbers $c_{i,m}$ count the number of times the integer $i$ appears as a factor in the different product representations of the integer $im$ (counting multiplicity). We define $c_{1,1}=0$. For instance, $k_{1}(y)$ is given by: $$k_{1}(y)=y^{2}+y^{3}+3y^{4}+y^{5}+3y^{6}+y^{7}+6y^{8}+3y^{9}+3y^{10}+.....$$ $k_{2}(y)$ is given by: $$y^{2}+2y^{4}+y^{6}+4y^{8}+y^{10}+3y^{12}+....$$ Now, suppose we want to write $k_{i}(y)$ as a Lambert series: $$k_{i}(y)=\sum_{n=1}^{\infty}\frac{b_{i,n}y^{in}}{1-y^{in}}$$ Where : $$c_{i,m}=\sum_{n|m}b_{i,n}$$ Or: $$b_{i,m}=\sum_{n|m}\mu(n)c_{i,\frac{m}{n}}$$ My questions:
1-What's the meaning/significance of the numbers $b_{i,m}$?
2-For $i=1$, and by induction, we have: $$b_{1,m}=\binom{\Omega(m)}{\omega(m)}$$ Where $\Omega(m)$ is the number of not necessarily distinct prime factors of $m$, and $\omega(m)$ is the number of distinct prime factors of $m$. How can we prove this rigorously ?