Oscillations in the preexponential of the nucleation rate are due to the discrete nature of small nuclei. An accurate elementary expression to describe such oscillations is derived in the limit of a high nucleation barrier. The result is applied to the standard Becker-Döring equation in two and three dimensions, and to the lowest-energy nucleation path in a cold lattice gas with Glauber and Metropolis dynamics (equivalent to an Ising model on a square lattice) where oscillation effects can be more pronounced.