For an integer $n>0$, we may define a polynomial $$p_n(x) = \sum_{d\mid n} x^d.$$Have such polynomials been seriously studied and do they have any interesting properties?
Looking at this more, I realize this might viewed as a $q$-analogue of the divisor counting function, but I am not too familiar with $q$-analogues and how they are used, so any information on this would be appreciated. As on outsider to this field, I assume that one often does not consider subtracting and multiplying $q$-analogues, and finding their magnitudes, so this may not be suitable for my purposes described. Regardless I am interested to hear any comments on this perspective if this is a studied $q$-analogue.
Motivation:
For distinct finite sets $A,B\subset \Bbb{N}$, there exists some minimal $k\le \min_{S\in\{A,B\}}\{\max_{x\in S}\{x\}\}$ such that $|\{n\in A:k\mid n\}|\neq |\{m\in B: k\mid m\}|$. We have that $$ \prod_{n\in A} p_n(x) -\prod_{n\in B} p_m(x) =\Theta(x^{k+L-1})$$near zero. (here $L = \min\{|A|,|B|\}$)
Right now this seems like a daunting way to calculate $k$ but I am wondering if these polynomials are well understood to perhaps inspire some way to use them in a proof.