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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 ?

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Could you please explain, with some examples, what you mean by "the number of times the integer $i$ appears as a factor in the different product representations of the integer $im$ (counting multiplicity)"? For example, could you list the product representations of 8 that give you 6 occurrences of the number 1, and the product representations of 12 that give you 3 occurrences of the number 2, and a few more examples? I know that 8 can be written as any of the products $1\cdot8,1\cdot2\cdot4,1\cdot2\cdot2\cdot2$, and I see 3 1's and 4 2's here, not 6 1's and 4 2's as your series seem to say. – Steve Kass Nov 2 '13 at 19:10
maybe the case where $i=1$ is a bit misleading. but, as you stated : $8=2.4=2.2.2$, or $8.1=$. so maybe i should've said -in this case- $c_{1,m}$ counts the factors of $m$ in each product representation. in the case where $i=2$ , and $m=12$, we have: $12=2.6=3.4=3.2.2$, so 2 appears as a factor 3 times in the different representations of $12$. – Mohammad Al Jamal Nov 2 '13 at 19:23
Mohammad, 8 is also equal to $1\cdot2\cdot1\cdot1\cdot2\cdot2$ and $2\cdot1\cdot2\cdot1\cdot2$. How are you deciding when and where factors of 1 are included in the products you write down? – Steve Kass Nov 2 '13 at 19:31
exactly !! that's why i said it is misleading in this case ! but in order for the definition of $c_{i,m}$ to be consistent we have to pick a convention ! i am still struggling with a better way to define $c_{i,m}$. Do you have any suggestions !? – Mohammad Al Jamal Nov 2 '13 at 19:33
Sorry, I don't have any suggestions, because I don't know what you're counting! If you provided more examples, I might have a guess, at least... – Steve Kass Nov 2 '13 at 19:40

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