We study a simple heteropolymer model containing sequence-independent local interactions on both square and triangular lattices. Sticking to a two-letter code, we investigate the model for varying strength kappa of the local interactions; kappa=0 corresponds to the well-known HP model [K. F. Lau and K. A. Dill, Macromolecules 22, 3986 (1989)]. By exhaustive enumerations for short chains, we obtain all structures which act as a unique and pronounced energy minimum for at least one sequence. We find that the number of such designable structures depends strongly on kappa. Also, we find that the number of designable structures can differ widely for the two lattices at a given kappa. This is the case, for example, at kappa=0, which implies that the HP model exhibits different behavior on the two lattices. Our findings clearly show that sequence-independent local properties of the chains can play an important role in the formation of unique minimum energy structures.