Collatz and company

Conjecture: The Collatz conjecture is just a disguised way of asking for a formula for the n-th prime, and will never be solved. You could make up other disguised ways of asking or a formula for the n-th prime. For example: let w be a positive integer. Then, let x(1) = w, and for each positive integer n > 1, if x(n – 1) = 1, stop. Otherwise, if x(n – 1) is square-free, let x(n) be the sum of the exponents in the prime factorization of x(n – 1), and if x(n – 1) is not square-free, let x(n) be 1 + x(n – 1). Then: Does this process stop for each and every starting positive integer w?
keywords: Mathematics, Number Theory