  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.
