## Product-sum numbers

### Problem 88

Published on Friday, 4th February 2005, 06:00 pm; Solved by 5396A 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}, ... , *a*_{k}} is called a product-sum number: N = *a*_{1} + *a*_{2} + ... + *a*_{k} = *a*_{1} × *a*_{2} × ... × *a*_{k}.

For example, 6 = 1 + 2 + 3 = 1 × 2 × 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 × 2 = 2 + 2

*k*=3: 6 = 1 × 2 × 3 = 1 + 2 + 3

*k*=4: 8 = 1 × 1 × 2 × 4 = 1 + 1 + 2 + 4

*k*=5: 8 = 1 × 1 × 2 × 2 × 2 = 1 + 1 + 2 + 2 + 2

*k*=6: 12 = 1 × 1 × 1 × 1 × 2 × 6 = 1 + 1 + 1 + 1 + 2 + 6

Hence for 2≤*k*≤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≤*k*≤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≤*k*≤12000?