I am starting to understand why this works. But starting from Euler's, how did you get here? And why is it that we don't have to check if i is prime?

It should be enough, and this approach would be expected to pass. I think you still did something wrong.
Sharing your code or asking at solving help channel could help.

This is awesome. Every time I submit there's something like this that inspires me to keep learning.

The way I understand this, it does.