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Denote by $S_n$ the group of permutations of the set $\{1,\ldots,n\}$ with composition as binary operation. Let $m_n$ denote the maximum order that an element of $S_n$ can have. What is the smallest positive integer $k$ such that $\lim_{n\to\infty} \frac{m_n}{n^k} < \infty$?

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    $\begingroup$ A simple OEIS check yields oeis.org/A000793, where also your question is answered as $lim_{n->\infty} (\log a(n)) / \sqrt{n \log n} = 1$. $\endgroup$ – Christian Stump Jan 25 at 12:05
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    $\begingroup$ From this post on Mathematics Maximal order of an element in a symmetric group we can learn that this is Landau's function. $\endgroup$ – Martin Sleziak Jan 25 at 12:06
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    $\begingroup$ This is Landau's function. There is no such $k$. $\endgroup$ – Derek Holt Jan 25 at 12:09
  • $\begingroup$ How strange! The question gets three downvotes while the (brief and correct) answer gets 10 upvotes. $\endgroup$ – Derek Holt Jan 26 at 15:57
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Landau proved in 1902 that the maximal order of an element in $S_n$ is $e^{(1+o(1))\sqrt{n\log n}}$. In particular, there is no integer $k$ with the property you ask for.

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