On pseudo-polynomials divisible only by a sparse set of primes and $\text{a}$-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 $p\mathrm{\mid }f(n)$ 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 $O(e^{(2\mathrm{/}3-ϵ)n})$, then it is a polynomial.