On pseudo-polynomials divisible only by a sparse set of primes and $\a$-primary pseudo-polynomials

We explore two questions about pseudo-polynomials, which are functions $f:\mathbb N \to \mathbb Z$ such that $k$ divides $f(n+k) - f(n)$ for all $n,k$. First, for certain arbitrarily sparse sets $R$, we construct pseudo-polynomials $f$ with $p|f(n)$ for some $n$ only if $p \in R$. This implies that not all pseudo-polynomials satisfy an assumption of a recent paper of Kowalski and Soundararajan. We also consider $\alpha$-primary pseudo-polynomials, where the pseudo-polynomial condition is only required for $k$ lying in a set of primes of density $\alpha$. We show that if an $\alpha$-primary pseudo-polynomial is $O(e^{(2/3-\epsilon) n})$, then it is a polynomial.