Counting Finite Index Subrings of ${\mathbb{Z}}^n$

We count subrings of small index of ${\mathbb{Z}}^n$, where the addition and multiplication are defined componentwise. Let $f_n(k)$ denote the number of subrings of index $k$. For any $n$, we give a formula for this quantity for all integers $k$ that are not divisible by a 9th power of a prime, extending a result of Liu.