Landau's function

From Academic Kids

Landau's function g(n) is defined for every natural number n to be the largest order of an element of the symmetric group Sn. 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 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000793).

The sequence is named after Edmund Landau, who proved that

<math>\lim_{n\to\infty}\frac{\ln(g(n))}{\sqrt{n \ln(n)}} = 1<math>

(where ln denotes the natural logarithm).

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

<math>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.<math>

External links

On-Line Encyclopedia of Integer Sequences: Sequence A000793, (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A000793) Landau's function on the natural numbers.

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