Problem 88


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, {a1, a2, … , ak} is called a product-sum number: N = a1 + a2 + … + ak = a1 × a2 × … × 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 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?


积和数

若自然数N能够同时表示成一组至少两个自然数{a1, a2, … , ak}的积和和,也即N = a1 + a2 + … + ak = a1 × a2 × … × ak,则N被称为积和数。

例如,6是积和数,因为6 = 1 + 2 + 3 = 1 × 2 × 3。

给定集合的规模k,我们称满足上述性质的最小N值为最小积和数。当k = 2、3、4、5、6时,最小积和数如下所示:

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

因此,对于2≤k≤6,所有的最小积和数的和为4+6+8+12 = 30;注意8只被计算了一次。

已知对于2≤k≤12,所有最小积和数构成的集合是{4, 6, 8, 12, 15, 16},这些数的和是61。

对于2≤k≤12000,所有最小积和数的和是多少?