Euler's Pentagonal Numbers Theorem/Corollary 1

Theorem

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

Let $\map {\sigma_1} n$ denote the divisor sum of $n$.

Then:

$\map {\sigma_1} n = \ds \sum_{1 \mathop \le n - GP_k \mathop < n} -\paren {-1}^{\ceiling {k / 2} } \map {\sigma_1} {n - GP_k} + n \sqbrk {\exists k \in \Z: GP_k = n}$

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

This theorem requires a proof. In particular: Follows somehow from Euler's Pentagonal Numbers Theorem, but at this time of night I have not a clue how. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Examples

${\sigma_1}$ of $12$

$\map {\sigma_1} {12} = \map {\sigma_1} {11} + \map {\sigma_1} {10} - \map {\sigma_1} 7 - \map {\sigma_1} 5 + 12$

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

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $22$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $22$