An observation on the sums of divisors

Abstract

Translation from the Latin of Euler's "Observatio de summis divisorum" (1752). E243 in the Enestroem index. The pentagonal number theorem is that $\prod_{n=1}^\infty (1-x^n)=\sum_{n=-\infty}^\infty (-1)^n x^{n(3n-1)/2}$. This paper assumes the pentagonal number theorem and uses it to prove a recurrence relation for the sum of divisors function. The term "pentagonal numbers" comes from polygonal numbers. Euler takes the logarithmic derivative of both sides. Then after multiplying both sides by $-x$, the left side is equal to $\sum_{n=1}^\infty \sigma(n) x^n$, where $\sigma(n)$ is the sum of the divisors of $n$, e.g. $\sigma(6)=12$. This then leads to the recurrence relation for $\sigma(n)$. I have been studying in detail all of Euler's work on the pentagonal number theorem, and more generally infinite products. I would be particularly interested to see if anyone else worked with products and series like these between Euler and Jacobi, and I would enjoy hearing from anyone who knows something about this.