Euler's Pentagonal Numbers Theorem/Corollary 1

Theorem
Let $n \in \Z_{>0}$ be a strictly positive integer.

Let $\map \sigma n$ denote the $\sigma$ function on $n$.

Then:
 * $\map \sigma n = \displaystyle \sum_{1 \mathop \le n - GP_k \mathop < n} -\paren {-1}^{\ceiling {k / 2} } \map \sigma {n - GP_k} + n \sqbrk {\exists k \in \Z: GP_k = n}$

where:
 * $\ceiling {k / 2}$ denotes the ceiling of $k / 2$
 * $GP_n$ denotes the $n$th generalized pentagonal number
 * $\sqbrk {\exists k \in \Z: GP_k = n}$ is Iverson's convention.