Let $F(n;i)$ be the number of labeled $i$-edge forests on $n$ vertices ([A138464][1] on the OEIS).  The first few values of $F(n;i) \pmod n$ are listed below:

$$\begin{array}{r|rrrrrrrrrrr}
  & i=0 & 1 & 2 & 3 & 4 & 6 & 7 & 8 & 9 & 10 & 11 \\
\hline
n=2 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
4 & 1 & 2 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
5 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
6 & 1 & 3 & 3 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
7 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
8 & 1 & 4 & 2 & 4 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
9 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
10 & 1 & 5 & 0 & 0 & 5 & 5 & 0 & 0 & 0 & 0 & 0 \\
11 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\end{array}$$

We see that if $n$ is an odd prime power and $i \geq 1$, then $n$ divides $F(n;i)$.  I can prove this via group actions and induction.

*Question*: Is there is a published proof of this result?

(Or, alternatively, a succinct proof of this result.)

  [1]: http://oeis.org/A138464