Problem 88

Product-sum Numbers

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: $N = a_{1} + a_{2} + \dots + a_{k} = a_{1} \times a_{2} \times \dots \times a_{k}$ .

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.

$\begin{aligned} k = 2 & : 4 = 2 \times 2 = 2 + 2 \\ k = 3 & : 6 = 1 \times 2 \times 3 = 1 + 2 + 3 \\ k = 4 & : 8 = 1 \times 1 \times 2 \times 4 = 1 + 1 + 2 + 4 \\ k = 5 & : 8 = 1 \times 1 \times 2 \times 2 \times 2 = 1 + 1 + 2 + 2 + 2 \\ k = 6 & : 12 = 1 \times 1 \times 1 \times 1 \times 2 \times 6 = 1 + 1 + 1 + 1 + 2 + 6 \end{aligned}$

Hence for $2 \leq k \leq 6$ , 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 $2 \leq k \leq 12$ is ${4, 6, 8, 12, 15, 16}$ , the sum is $61$ .

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

积和数

若自然数 $N$ 能够同时表示成一组至少两个自然数 ${a_{1}, a_{2}, \dots, a_{k}}$ 的积与和，也即 $N = a_{1} + a_{2} + \dots + a_{k} = a_{1} \times a_{2} \times \dots \times a_{k}$ ，则称之为积和数。

例如， $6$ 是积和数，因为 $6 = 1 + 2 + 3 = 1 \times 2 \times 3$ 。

给定这一组自然数的数目 $k$ ，满足上述性质的最小 $N$ 值被称为最小积和数。当 $k = 2$ 、 $3$ 、 $4$ 、 $5$ 、 $6$ 时，最小积和数如下所示：

因此，对于 $2 \leq k \leq 6$ ，所有的最小积和数之和为 $4 + 6 + 8 + 12 = 30$ ；注意 $8$ 只被计算了一次。

已知对于 $2 \leq k \leq 12$ ，所有最小积和数构成的集合是 ${4, 6, 8, 12, 15, 16}$ ，这些数之和是 $61$ 。

对于 $2 \leq k \leq 12000$ ，所有最小积和数之和是多少？