[function_seven]

> Trait A is that the numbers have a certain kind of similarity in their spacing, leading to few unique sums. Trait B is that they have a certain kind of similarity in their factors, leading to few unique products.

This is an excellent way to describe it. I had the sense that “of course the geometric sequences will collide more when multiplying” but if you asked me “why?” I would have trouble explaining. So thanks.

As far as finding a set that has an equal number of distinct sums and products, would it be possible to come up with a sequence that’s “half”-arithmetic and “half”-geometric?

[Dylan16807]

Even a single value that doesn't fit in one sequence will cause a lot of extra values, so your ability to fit multiple patterns gets worse and worse as you go longer. Something like 2 4 6 8 16 does pretty well for its size but even then it's giving you twelve distinct outputs for either operation.