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\le k\le n\Big\}& \mbox{ if } n>0\end{matrix}\right.

External links

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

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