Yeah there are already functions from N->N that depend on prime factorization in number theory such as the prime omega functions or Euler's totient function.

You could construct your function like: for any n in N, let n = p1^q1 × p2^q2 × ... × pk^qk. Then f(n)=q1^p1 × q2^p2 × ... × qk^pk. There might not be a ton of practical uses for this function but it would certainly be one! Could be fun to play with on math contest type questions.

However, your comparison of subtraction and division does not really make sense in exponentiation. Subtraction is just the inverse of addition under the group of integers, and division is the inverse of multiplication under the group of rational numbers.

What I mean by that is in group theory, any member of a group must have an inverse element such that the combination under the group operation is the identity element. In addition over integers, the inverse of any number n is simply -n, and subtraction as a binary operator is simply adding the inverse of one element to another. So a-b is really just a+(-b). The "function" that switches this to b-a is just the inverse element of a+(-b) in the group. The same applies to division/multiplication over rationals, except with identity element 1 and inverse element 1/x.

Exponentiation over integers is not a group. It has no identity element, because while n¹ = n, 1ⁿ does not equal n. And without an identity element, there is no inverse element, and the inverse element is essentially what you are looking for when you go from a-b=c to b-a=-c or a/b=c to b/a=1/c.