One mathematics question

First of all, every element in the group Sn can be written as disjoint cycles
(.....) (...) (.......)
Let a1, a2..., ak be the size of these cycles. Then a1+a2+...+ak=n

Now what is the order of this element? It is the LCM of a1,a2...,ak
and LCM(a1,a2,...,ak) <= a1 a2... ak

We will show that a1 a2... ak < e^(n/e)

Since there are k numbers adding to n, a1+a2+...+ak=n,
by AM-GM inequality, we get a1 a2... ak <= (n/k)^k

Now, by checking the derivatives with respect to k, we see that
d/dk (n/k)^k = (n/k)^k (log n/k - 1)
so that it has maximum value at n/k = e, or k = n/e

So our conclusion is: order of element <= a1 a2... ak <= (n/k)^k < e^(n/e)

en/e= (e^(1/e))^n ≈ 1.44n (By calculating the value of (e)^(1/e))
e = 2.718... (e^(1/e)) = 1.444667861