I was hoping someone might know the name for this representation of the positive integers. The idea is similar to Peano's arithmetic but applied to multiplication. So we have $1$, multiplication and a function $p$. We map onto the normal integers by having p map to the nth prime. To illustrate the first few integers would be $1, p(1), p(p(1)), p(1)p(1), p(p(p(1)))$.
This representation has some unusual properties. Computing whether a number is prime is trivial, as is factorization and multiplication.
Addition however is rather tricky since you have to convert back to a normal representation, perform the addition then recursively factorized the result!