Euler's Pentagonal Numbers Theorem/Corollary 1

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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.


Proof


Examples

$\sigma$ of $12$

$\map \sigma {12} = \map \sigma {11} + \map \sigma {10} - \map \sigma 7 - \map \sigma 5 + 12$


Historical Note

Leonhard Paul Euler had noticed, but had not formally proved, what is now known as Euler's Pentagonal Numbers Theorem.

However, he was so convinced of its truth that he used it to prove this and other results.

Hence he demonstrated the interesting fact that, in order to calculate the sum of the divisors of a number, it is necessary merely to know the sum of the divisors of the relevant smaller numbers, but not what those divisors themselves actually are.


Sources