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!).

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