Number of abelian groups of order $p^n$

Your work is correct, except that you aren't answering the question asked (they asked you for the number of (nonisomorphic) groups, not for a list of the groups). So for $n=1$, the answer should be "1"; for $n=2$ the answer should be "2"; for $n=3$ the answer should be "3"; for $n=4$ the answer should be "5", etc.

The magic words you are looking for are "partitions of $n$." You should verify that there is a bijection between the isomorphism types of abelian groups of order $p^n$ and the partitions of $n$.

It's equal to the number of partitions of $n$. See http://mathworld.wolfram.com/AbelianGroup.html