  Let ${\mathfrak S}_n$ denote the symmetric group with $n$ letters, and $g(n)$
the maximal order of an element of ${\mathfrak S}_n$. If the standard
factorization of $M$ into primes is $M=q_1^{\al_1}q_2^{\al_2}... q_k^{\al_k}$,
we define $\ell(M)$ to be $q_1^{\al_1}+q_2^{\al_2}+... +q_k^{\al_k}$; one
century ago, E. Landau proved that $g(n)=\max_{\ell(M)\le n} M$ and that, when
$n$ goes to infinity, $\log g(n) \sim \sqrt{n\log(n)}$. There exists a basic
algorithm to compute $g(n)$ for $1 \le n \le N$; its running time is
$\co(N^{3/2}/\sqrt{\log N})$ and the needed memory is $\co(N)$; it allows
computing $g(n)$ up to, say, one million. We describe an algorithm to calculate
$g(n)$ for $n$ up to $10^{15}$. The main idea is to use the so-called {\it
$\ell$-superchampion numbers}. Similar numbers, the {\it superior highly
composite numbers}, were introduced by S. Ramanujan to study large values of
the divisor function $\tau(n)=\sum_{d\dv n} 1$.

Number Theory