A natural number, N, that can be written as the sum and product of a given set of at least two natural numbers, {a1, a2, ... , ak} is called a product-sum number: N = a1 + a2 + ... + ak = a1a2
...
ak.
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 2k
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 2k
12 is {4, 6, 8, 12, 15, 16}, the sum is 61.
What is the sum of all the minimal product-sum numbers for 2k
12000?
These problems are part of Project Euler and are licensed under CC BY-NC-SA 2.0 UK