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 all partitions of the numbers 1 to . 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 showed that
Local maxima of this function 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.
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.
Monthly109, 173-186, 2002.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. France97, 129-191,
1969.Nicolas, J.-L. "On Landau's Function ." In The
Mathematics of Paul Erdos: Part 1 (Ed. R. L. Graham et al.).
pp. 228-240.Sloane, N. J. A. Sequences A000793/M0537,
A103635, and A129759
in "The On-Line Encyclopedia of Integer Sequences."
is the maximum order of an element in the symmetric
group 
.
The value 
is given by the largest least common multiple
of all partitions of the numbers 1 to 
. 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).
![lim_(n->infty)(ln[g(n)])/(sqrt(nlnn))=1.](/images/equations/LandausFunction/NumberedEquation1.svg)
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.
." Math. Comput. 64, 407-410, 1995.Haack,
J. "The Mathematics of Steve Reich's Clapping Music." In 
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