Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $\mathbb{C}$).
Given a partition $p=[a\_1,...,a\_m]$ of an integer $n$. We can identify $p$ with the semisimple algebra $A\_p:=M\_{a\_1}(K) \times \cdots \times M\_{a\_m}(K)$.

Questions:

1. Given a partition $p$, which finite groups $G$ have their group algebra over $K$ being isomorphic to $A\_p$?
2. Given a natural number $n$, how many partitions p with sum $n$ are there such that $A\_p$ is isomorphic to a group algebra?
3. How does the sequence of numbers of such partitions begin depending on $K$ (Does it in general depend on the field or perhaps just the characteristic of the field?)?

Probably the answer is very complicated, but can something be said about the rough behavior of the sequence?

It should start as follows for $n \geq 1$ and $K=\mathbb{C}$, using GAP: 1,1,1,2,1,4,1,4,2,4,1,9,1,5,4

4. $\textbf{Are those numbers always powers of primes?}$. (The next term takes forever to calculate, but maybe I should wait longer before asking this question....)

I dont know how complicated those questions are, so partial answers are also welcome.