_{This problem is a programming version of Problem 88 from projecteuler.net}

A natural number, , that can be written as the sum and product of a given set of at least two natural numbers, is called a product-sum number: .

For example, .

For a given set of size, , we shall call the smallest with this property a minimal product-sum number. The minimal product-sum numbers for sets of size, are as follows.

Hence for , the sum of all the minimal product-sum numbers is ; note that is only counted once in the sum.

In fact, as the complete set of minimal product-sum numbers for is , the sum is .

What is the sum of all the minimal product-sum numbers for ?