Problem 159
Digital root sums of factorisations
A composite number can be factored many different ways. For instance, not including multiplication by one, 24 can be factored in 7 distinct ways:
24 = 2x2x2x3
24 = 2x3x4
24 = 2x2x6
24 = 4x6
24 = 3x8
24 = 2x12
24 = 24
Recall that the digital root of a number, in base 10, is found by adding together the digits of that number, and repeating that process until a number is arrived at that is less than 10. Thus the digital root of 467 is 8.
We shall call a Digital Root Sum (DRS) the sum of the digital roots of the individual factors of our number.
The chart below demonstrates all of the DRS values for 24.
| Factorisation | Digital Root Sum |
|---|---|
| 2x2x2x3 | 9 |
| 2x3x4 | 9 |
| 2x2x6 | 10 |
| 4x6 | 10 |
| 3x8 | 11 |
| 2x12 | 5 |
| 24 | 6 |
The maximum Digital Root Sum of 24 is 11.
The function mdrs(n) gives the maximum Digital Root Sum of n. So mdrs(24)=11.
Find ∑mdrs(n) for 1 < n < 1,000,000.
分解约数数字根之和
每个合数都有很多种分解约数的方式。例如，如果不允许多次乘1，有7种不同的方式分解24的约数：
24 = 2x2x2x3
24 = 2x3x4
24 = 2x2x6
24 = 4x6
24 = 3x8
24 = 2x12
24 = 24
在十进制下，一个数的数字根是不断重复将其各位数字相加，直到得到的结果小于10为止。因此467的数字根是8。
我们记数字根和（DRS）是所有分解出的约数的数字根之和。
下表是24的所有约数分解的DRS值。
| 约数分解 | 数字根和 |
|---|---|
| 2x2x2x3 | 9 |
| 2x3x4 | 9 |
| 2x2x6 | 10 |
| 4x6 | 10 |
| 3x8 | 11 |
| 2x12 | 5 |
| 24 | 6 |
对于24的所有分解，最大的数字根和是11。
函数mdrs(n)表示对于n的不同分解最大的数字根和。因此mdrs(24)=11。
对于1 < n < 1,000,000，求∑mdrs(n)。