Eccoci.

A natural number, $N$, that can be written as the sum and product of a given set of at least two natural numbers, $\{a_1, a_2, \dots, a_k\}$ is called a product-sum number: $N = a_1 + a_2 + \cdots + a_k = a_1 \times a_2 \times \cdots \times a_k$.

For example, $6 = 1 + 2 + 3 = 1 \times 2 \times 3$.

For a given set of size, $k$, we shall call the smallest $N$ with this property a minimal product-sum number. The minimal product-sum numbers for sets of size, $k = 2, 3, 4, 5$, and $6$ are as follows.

• $k=2$: $4 = 2 \times 2 = 2 + 2$
• $k=3$: $6 = 1 \times 2 \times 3 = 1 + 2 + 3$
• $k=4$: $8 = 1 \times 1 \times 2 \times 4 = 1 + 1 + 2 + 4$
• $k=5$: $8 = 1 \times 1 \times 2 \times 2 \times 2 = 1 + 1 + 2 + 2 + 2$
• $k=6$: $12 = 1 \times 1 \times 1 \times 1 \times 2 \times 6 = 1 + 1 + 1 + 1 + 2 + 6$

Hence for $2 \le k \le 6$, the sum of all the minimal product-sum numbers is $4+6+8+12 = 30$; note that $8$ is only counted once in the sum.

In fact, as the complete set of minimal product-sum numbers for $2 \le k \le 12$ is $\{4, 6, 8, 12, 15, 16\}$, the sum is $61$.

What is the sum of all the minimal product-sum numbers for $2 \le k \le 12000$?