## Factorisation triples

### Problem 418

Published on Saturday, 9th March 2013, 07:00 pm; Solved by 407; Difficulty rating: 40%
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!).