Define a formal power series like so: $$P_2(x) = \frac{x^2}{1-x^2} + \frac{x^4}{1-x^4}+\frac{x^8}{1-x^8}+\cdots$$
I rigged the definition of $P_2$ so that the sequence of coefficients of $x^i$ looks like this:
$$0,1,0,2,0,1,0,3,0,1,0,2,\ldots$$
This sequence comes up in number theory. If we ask how many factors of $2$ occur in the numbers $1,2,3, \ldots$, we obtain the above sequence.
More generally, define $$P_p(x) = \sum_{i=1}^\infty \frac{x^{p^i}}{1-x^{p^i}}.$$
If I'm not mistaken, the corresponding sequence of coefficients arise as the number of factors of $p$ in the numbers $1,2,3,4, \ldots$.
Question. What are these formal power series really called?
Remark. My original question was edited under the mistaken premise that these are not formal power series. In fact, they are. For instance, since $1-x^2$ is a non-zero formal power series, hence so too is $\frac{1}{1-x^2},$ which is by definition the unique $Q \in \mathbb{R}(x)$ such that $Q(1-x^2) = 1$, which turns out to be $$Q = 1+x^2+x^4+\cdots.$$ Furthermore, $\mathbb{R}(x)$ carries a topology, which allows us to take limits; under this viewpoint $P_2(x)$ is a limit of partial sums, and this sequence of partial sums is convergence, $P_2(x)$ is a perfectly good formal power series. And, in future, please don't edit my questions to reduce interpersonal conflict. The OP can edit their own question themselves if they realize they're wrong. And if they're not wrong, I'd much prefer that he/she stands their ground aggressively, which is, after all, much more admirable than backing down when you're right.
I'm not sure it has a specific name, it's basically the ordinary generating function of the 2-adic valuation, with the only difference that you take $x^{-n}$ in the formula rather than $x^n$ (this is sometimes done for convenience).
If you want a name, I'd go with $2$-adic (ordinary) generating function. I don't think it's got any well-established name.