Polya cites this work of Euler as an example of a conjecture which Euler considered impossible to doubt, and yet still needing a demonstration.
It is on pages 90-98 of "Induction and Analogy in Mathematics", which is volume I of "Mathematics and Plausible Reasoning", Princeton 1954. The reference to Euler is in a footnote on page 90. (Opera Omnia ser 1 vol 2 p 241-253).
The conjecture is a recursion formula for the sum of the divisors of any integer, it is stated on page 93 of Polya.
Thanks for any help.
Edit: In response to comments, here is the conjecture. For $n \geq 1$, let $\sigma(n) = \sum_{d|n} d$ be the sum of divisors of $n$. Euler conjectures the remarkable formula
$$\begin{align*} &\ \sigma(n)\\ =&\ \sigma(n-1) + \sigma(n-2) - \sigma(n-5) - \sigma(n-7) + \sigma(n-12) + \sigma(n-15) - \sigma(n-22) + \ldots \\ =&\ \sum_{k \geq 1} (-1)^{\text{floor}\left(\frac{k-1}{2}\right)} \sigma(n - f(k)) \end{align*} $$
where $f(k)$ is defined recursively by $f(1) = 1$ and $f(k) = f(k-1) + \frac{k}{2}$ if $k$ is even, $f(k) = f(k-1) + k$ if $k$ is odd, for all $k > 1$. (Euler stipulates that $\sigma(n - f(k)) := 0$ if $n - f(k) < 0$, and one should substitute $n$ for $\sigma(0)$ if such a summand should arise on the right.)
My question is whether this has ever been proved, and if so where I could find the proof.