Newton representation of functions over natural integers having integral difference ratios

Different questions lead to the same class of functions from natural integers to integers: those which have integral difference ratios, i.e. verifying f(a)− f(b) ≡ 0 (mod (a− b)) for all a > b. We characterize this class of functions via their representations as Newton series. This class, which obviously contains all polynomials with integral coefficients, also contains unexpected functions, for instance all functions x 7→ be a x!c, with a ∈ Z \ {0, 1}, and a function equal to be x!c except on 0. Finally, to study the complement class, we look at functions N → R which are not uniformly close to any function having integral difference ratios.