Problem Archive

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

• $1 \leq a \leq b \leq c$
• $a \cdot b \cdot c = n$.

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 $f(43!)$.