i dont get it...what is the hidden connection exactly?
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.
So from some angle it's interesting that you can't impose both types of patterns on a single set of numbers, because that inability implies a relation between two very different operations.
But on the other hand why would you assume that you should find an intersection between two restrictive algorithms? Maybe there's a hidden assumption that the algorithms would be random-ish and let you find an overlap if you search hard enough, and that assumption is screwing with people's intuition.
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?
I think it's key here that these two algorithms are so basic and fundamental to the structure they are executed upon (the naturals) so that every other algorithm will use them in some way. The very notion of "algorithm over the naturals" and intuition of what it means for such an algorithm to be restrictive are determined by the nature of these operations.
Put like this, it seems far less obvious to me that there is an a priori reason why the results of these two operations should or shouldn't coincide. Both scenarios seem plausible before we study problems like these.