In this paper and its sequel we give results and methods for evaluating the Nielsen type numbersN Φ n (f) and NP n (f) for periodic points of self maps f of nilmanifolds and solvmanifolds (which includes the Klein bottle). Using fibre space techniques we relate these numbers to the various ordinary Nielsen numbers N(f m ) of the mth iterate (for m n), and thus capitalize on known techniques for our calculations. In this first paper we deal with the specific situation where for each m n either N(f m ) = R(f m ) (the Reidemeister number off m ) or N(f m ) = 0. In fact this is the vast majority of cases. In the case where N(f n ) = R(f n ) we derive the simple formulaN Φ m (f) = N(f m ) for all m n.