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

We explore two questions about pseudo-polynomials, which are functions f : N → 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 ∈ R. This implies that not all pseudo-polynomials satisfy an assumption of a recent paper of Kowalski and Soundararajan. We also consider α-primary pseudo-polynomials, where the pseudo-polynomial condition is only required for k lying in a set of primes of density α. We show that if an α-primary pseudo-polynomial is O(e(β−)n), where β = √7 3 − 1 6 ≈ 0.715, then it is a polynomial.