I know that Beurling developed a notion of generalized primes (and integers.

However, does anyone know if Beurling, or anyone else, studied subclasses of the broader class of Beurling primes that are generated with specific rules?

For example, I have in mind a method of constructing generalized primes by starting with the natural numbers N (well, N+ really), then selecting an operation/function that is not just the usual multiplication, generating a set of generalized composites (C) by operating this function on all binary combos from N, and then taking the generalized primes to be P=N-C.

My thought is that if you could look at sequences of operations (generalizing multiplication for classical primes) that generate different sequences of primes and study how the properties of those sequences change as you "move away or towards" the primes (if you had some suitable way of parameterizing the sequences of operations).

My hunch is one or more of the following is true:

(1) Buerling's studies of generalized primes (or extensions of it) already treat such constructions, either implicitly or explicitly and so I can go learn about them in his works.

(2) These alternately constructed primes may bear little to no (useful) relation to the classical primes, or may not change in any "regular" way as you move towards the classical primes that would allow one to gain information about the classical primes.

(3) Most (or all?) alternately constructed primes may not be different enough from the primes, so that while specific scales of gaps between primes, etc. may change, their overall behavior may be rather similar, so again, not very useful for finding new information about the classical primes.

Does anyone know if (1), (2) or (3) are true, and/or have any guidance about specific topics to explore in relation to looking at specific collections/sequences of generalized primes?