TOPICS
Landau's Function
Landau's function 
is the maximum order of an element in the symmetric
group 
.
The value 
is given by the largest least common multiple
of the parts in a partition of 
. The first few values for 
, 2, ... are 1, 2, 3, 4, 6, 6, 12, 15, 20, 30, ... (OEIS
A000793), and have been computed up to 
by Grantham (1995).
Landau (1903) showed that
![lim_(n->infty)(ln[g(n)])/(sqrt(nlnn))=1.](https://mathworld.wolfram.com/images/equations/LandausFunction/NumberedEquation1.svg) |
Successive record values of ![ln[g(n)]/sqrt(nlnn)](https://mathworld.wolfram.com/images/equations/LandausFunction/Inline7.svg)
occur at 2, 3, 5, 7, 9, 10, 12, 17, 19,
30, 36, 40, ... (OEIS A103635).
Let 
be the greatest prime factor of 
. Then the first few terms for 
, 3, ... are 2, 3, 2, 3, 3, 3, 5, 5, 5, 5, 5, 5, 7, 7, 7,
7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 11, ... (OEIS A129759).
Nicolas (1969) showed that 
. Massias et al. (1988, 1989)
showed that for all 
,
, and Grantham
(1995) showed that for all 
, the constant 2.86 may be replaced by 1.328.
See also
Group Order, Symmetric GroupExplore with Wolfram|Alpha
References
Grantham, J. "The Largest Prime Dividing the Maximal Order of an Element of
." Math. Comput. 64, 407-410, 1995.Haack,
J. "The Mathematics of Steve Reich's Clapping Music." In Bridges:
Mathematical Connections in Art, Music, and Science: Conference Proceedings, 1998
(Ed. R. Sarhangi), pp. 87-92, 1998.Kuzmanovich, J. and Pavlichenkov,
A. "Finite Groups of Matrices Whose Entries Are Integers." Amer. Math.
Monthly 109, 173-186, 2002.Landau, E. "Über die
Maximalordnung der Permutationen gegebenen Grades." Arch. Math. Phys. Ser.
3 5, 92-103, 1903.Massias, J.-P. "Majoration explicite
de l'ordre maximum d'un élément du groupe symétrique."
Ann. Fac. Sci. Toulouse Math. 6, 269-281, 1984.Massias,
J.-P.; Nicolas, J.-L.; and Robin, G. "Évaluation asymptotique de l'ordre
maximum d'un élément du groupe symétrique." Acta Arith. 50,
221-242, 1988.Massias, J.-P.; Nicolas, J.-L.; and Robin, G. "Effective
Bounds for the Maximal Order of an Element in the Symmetric Group." Math.
Comput. 53, 665-678, 1989.Miller, W. "The Maximum Order
of an Element of a Finite Symmetric Group." Amer. Math. Monthly. 94,
497-506, 1987.Nicolas, J.-L. "Sur l'ordre maximum d'un élément
dans le groupe 
des permutations." Acta Arith. 14, 315-322, 1968.Nicolas,
J.-L. "Ordre maximum d'un élément du groupe de permutations et
highly composite numbers." Bull. Math. Soc. France 97, 129-191,
1969.Nicolas, J.-L. "On Landau's Function 
." In The
Mathematics of Paul Erdos: Part 1 (Ed. R. L. Graham and J. Nešetril).
pp. 228-240.Sloane, N. J. A. Sequences A000793/M0537,
A103635, and A129759
in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Landau's FunctionCite this as:
Weisstein, Eric W. "Landau's Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LandausFunction.html