number theory for finding value of $k$ How do I find what is the smallest positive integer $k$ such that $(3^3 + 4^3 + 5^3)\cdot k = a^n$ for some positive integers $a$ and $n$, with $n > 1$?

Since $3^3+4^3+5^3=6^3$, we have 
$$
6^3k=a^n
$$
Thus $2|a$ and $3|a$, so $6|a$.
Now for $n=1$, $k=1$ suffices, and it is easy to see that for $n=2$, $k=6$ is good, and no smaller number suffices. If $n=3$, then $k=1$ is good as well. Otherwise, $a\geq 6\Rightarrow$ $a^n\geq 6^n$ and thus $k\geq 6^{n-3}$. But $k=6^{n-3}$ is a good choice.
