@Hhan: thanks for your observation. I’m still trying to make sense of that paper but part of the issue seems that they only have an upper bound so I’m not sure if that would be enough to conclude what happens to my ratio.

@YCor: yes. Technically if $p^{[0]}(n)$ has positive integer zeros, then we need additional initial conditions to ensure uniqueness.

@NewBornMATH: math.lsa.umich.edu/~lagarias/doc/elementaryrh.pdf