Let $p$ be a prime, $n$ a non-negative integer. On one hand, the sum of the divisors of $p^n$ (denoted by $\sigma_1(p^n)$) is given by $\sigma_1(p^n)=1+ \dots +p^n=\cfrac{p^{n+1}-1}{p-1}$.
The number of 1 dimensional subspaces of $\mathbb{F}_p^{n+1}$ is computed by the number of nonzero vectors in $\mathbb{F}_p^{n+1}$, divided by the number of bases for a 1 dimensional subspace (this procedure can of course be generalized for the number of $k$-dimensional subspaces of $\mathbb{F}_p^{m}$), which yields the same formula given above, namely $\cfrac{p^{n+1}-1}{p-1}$.
My question is regarding these formulas. Why are they the same? Is it pure coincidence, or is there some connection I am not seeing?