A published proof for: the number of labeled $i$-edge ($i \geq 1$) forests on $p^k$ vertices is divisible by $p^k$ Let $F(n;i)$ be the number of labeled $i$-edge forests on $n$ vertices (A138464 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.)

The proof is now Lemma 2 here:

A. P. Mani, R. J. Stones, Congruences for the weighted number of labeled forests.  Integers, 16 (2016): A17.

which is freely available from: http://www.integers-ejcnt.org/vol16.html
