---

Problem 418

---

Factorisation triples

Let n be a positive integer. An integer triple (a, b, c) is called a factorisation triple of n if:

• 1 ≤ a ≤ b ≤ c
• a·b·c = n.

Define f(n) to be a + b + c for the factorisation triple (a, b, c) of n which minimises c / a. One can show that this triple is unique.

For example, f(165) = 19, f(100100) = 142 and f(20!) = 4034872.

Find f(43!).

---

因数三元组

记n是一个正整数。一个整数三元组(a, b, c)被称为n的因数三元组，当满足一下条件：

• 1 ≤ a ≤ b ≤ c
• a·b·c = n.

在n因数三元组(a, b, c)中，取使得c / a最小化的一组，记f(n)是这三个数的和a + b + c。可以验证这样的三元组总是唯一的。

例如f(165) = 19，f(100100) = 142，而f(20!) = 4034872。

求f(43!)。

---