Counting subrings of Zn of index k

We consider the problem of determining the number of subrings of the ring Zn of a fixed index k, denoted fn(k). We present a decomposition theorem for these subrings and calculate explicit expressions for the Dirichlet series generating function for n⩽4 and for the generating function modulo p. We also calculate fn(k) when k is not divisible by the sixth power of any prime.