A note on the partitions involving parts of k different magnitudes

The natural expression of the generating function Nk(q) for the number of partitions of the positive integer n that have exactly k distinct values for the parts was remarked by MacMahon. A factorization of this generating function was derived later by Andrews. In this paper, the author considers a truncated form of Nk(q) to give a new proof of Andrews's identity. A generalization of a recent connection between partitions and divisors is presented in this context.