In "Algebra" course by Vinberg (Винберг Э.Б.) there is the following problem (4.3.1 on p. 161 of Russian edition, marked as hard):
We know that every $\tau\in S_n$, where $S_n$ is a permutation group of $n$ elements, can be decomposed to the product of cycles: $\tau = p_1\dots p_k$. From there it is easy to see that the order of $\tau$ is $\mathrm{LCM}(|p_1|, \dots,|p_k|)$. The problem asks to prove that this order is less than
$$ e^{n/e} $$
So far I figured out that the maximum order of $\tau$ would be in the case when the order of each $p_i$ is a distinct prime number. Furthermore, the maximum would be if they are consecutive and small $2,3,5,\dots$ rather than big ones, since this way their product will be bigger. So, I am thinking now the problem can be re-stated as:
Let $n$ be a sum of the first $k$ prime numbers $n=2+3+5+\dots+p_k$, then I need to prove that
$$ 2\times3\times5\times\dots\times p_k < e^{n/e} $$
But I can't find the way to move forward. I can't think of any estimate of prime numbers product, let alone the one that connects it to $e$. Is my approach even correct?
PS. I found that what I am looking for is called Landau's function $g(n)$. The fact that $g(n)<e^{n/e}$ is stated in Wikipedia article, but no proof or citation is given.
PPS. One idea I've just thought is to take $\log$ of the product --- still can't find my way through it, but it seems I'm getting closer...