I'm not sure if this is easy or hard, but I have been wondering about this for a little while and can't get a solution.

Say that a factorization of a natural number n is a set multiset of natural numbers {a_1, a_2, ..., a_k}, all greater than 1, such that a_1 * a_2 *... * a_k = n.

For a given n, how many different factorizations are there?

Example: {2,2,3}, {4,3}, {6,2} and {12} are factorizations of 12. They are the only factorizations.

If n is of the form p^m, where p is prime, then the answer is the number of partitions of m. If n is a square free number, with m prime factors, then the answer is the m-th Bell number. Is there a general answer? Closed form, or maybe a recursive relationship?