Landau's function

Landau's function g(n) is defined for every natural number n to be the largest order of an element of the symmetric group S_n. Equivalently, g(n) is the largest least common multiple of any partition of n.

For instance, 5 = 2 + 3 and lcm(2,3) = 6. No other partition of 5 yields a bigger lcm, so g(5) = 6. An element of order 6 in the group S₅ can be written in cycle notation as (1 2) (3 4 5).

The integer sequence g(0) = 1, g(1) = 1, g(2) = 2, g(3) = 3, g(4) = 4, g(5) = 6, g(6) = 6, g(7) = 12, g(8) = 15, ... is A000793.

The sequence is named after Edmund Landau, who proved that

$\lim _{n\to \infty }{\frac {\ln(g(n))}{\sqrt {n\ln(n)}}}=1$

(where ln denotes the natural logarithm).

The following recursive formula can be used to compute g(n):

$g(n)=\left\{{\begin{matrix}1&{\mbox{ if }}n=0\\\max {\Big \{}\operatorname {lcm} (k,g(n-k))\mid 1\leq k\leq n{\Big \}}&{\mbox{ if }}n>0\end{matrix}}\right.$

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External Link:

On-Line Encyclopedia of Integer Sequences: Sequence A000793, Landau's function on the natural numbers.