Modeling Prime Factorization with Bijections between Natural Numbers, Finite Sequences and Multisets

We describe mechanisms to emulate properties of prime numbers by connecting them to computationally simpler representations involving bijections from natural numbers to sequences and multisets of natural numbers. Through a generic framework, implemented as a Haskell type class, we model interesting properties of canonical representation of factorization and its simpler equivalents, like information lost through factorization, a connection to Catalan families and similarity of behavior of the corresponding concepts of smooth numbers and the Ω and ω functions.