Please consider this a SOFT and very general question.
Ever since first being introduced to the Sieve of Eratosthenes (many decades ago) I have always visualized it as applying combs to the number line. This mental visualization does not entail a mathematical formalization, although highly convoluted and ad hoc formalizations can be conceived of. Rather, it reflects my own mental image of what is happening. I start with the first non-unit number, $2$, and make a comb with teeth spaced $2$ units apart, and lay the comb on the number line with the first tooth on $2^2$. Then I successively take the next smallest remaining number $n$, make a comb with teeth spaced $n$ units apart, and lay it on the number line with its first tooth on $n^2$. The numbers not covered by the teeth of any comb are the primes.
This is not at all a new result, just my idiosyncratic way of visualizing well known facts. It is no different in effect from serially crossing out or otherwise removing numbers that are multiples of primes. In operation, however, it is somewhat distinct in that it creates structures (combs) and applies them in a single operation to the number line, as opposed to testing divisibilities of each number, or progressing gap by gap along the number line, as seems implicit in the standard appreciation of the Sieve of Eratosthenes. Stated another way, the comb structure allows you to cross out all of a category of numbers at once (by virtue of their position), rather than crossing them out one at a time (by virtue of their divisibility properties). Perhaps this distinction will be considered overly subtle by some readers.
I have wondered if there is any formal mathematics that is based on this kind of visualization. The closest thing I can find concerns Dirac combs, but this is not particularly apt.
My question is: Are there mathematical formalisms whereby structures (like combs) are created and which are then used to identify or otherwise operate on members of a collection of numbers (or other objects) by virtue of their position, such as on a line or in an array? If so, where can I read about such mathematics?
I'd say your comb visualization is part of the "standard appreciation" of the sieve. The comb structure you're describing is just an infinite set.