# Difference between revisions of "Landau's function"

From Number

(Created page with '==Definition== Let <math>n</math> be a nonnegative integer. The '''Landau's function''' of <math>n</math> is defined in the following equivalent ways: * The maximum possible va...') |
|||

Line 4: | Line 4: | ||

* The maximum possible value for the [[least common multiple]] of the parts in an [[unordered integer partition]] of <math>n</math>. | * The maximum possible value for the [[least common multiple]] of the parts in an [[unordered integer partition]] of <math>n</math>. | ||

− | * The maximum of the [[groupprops:order of an element|order]]s of all elements in the symmetric group of degree <math>n</math>. | + | * The maximum of the [[groupprops:order of an element|order]]s of all elements in the [[groupprops:symmetric group|symmetric group]] of degree <math>n</math>. |

The value at <math>n = 0</math> is defined to be <math>1</math>. | The value at <math>n = 0</math> is defined to be <math>1</math>. |

## Latest revision as of 21:00, 30 April 2009

## Definition

Let be a nonnegative integer. The **Landau's function** of is defined in the following equivalent ways:

- The maximum possible value for the least common multiple of the parts in an unordered integer partition of .
- The maximum of the orders of all elements in the symmetric group of degree .

The value at is defined to be .

## Behavior

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A000793

## =Initial values

The values of Landau's function for is .

## Relation with other functions

- lcm of all numbers so far is the exponent of the symmetric group of degree .
- factorial is the order (i.e., cardinality) of the symmetric group of degree .