Landau's Function


Landau's function g(n) is the maximum order of an element in the symmetric group S_n. The value g(n) is given by the largest least common multiple of all partitions of the numbers 1 to n. The first few values for n=1, 2, ... are 1, 2, 3, 4, 6, 6, 12, 15, 20, 30, ... (OEIS A000793), and have been computed up to n=500000 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 gpf(a(n)) be the greatest prime factor of g(n). Then the first few terms for n=2, 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 gpf(g(n)))∼sqrt(nlnn). Massias et al. (1988, 1989) showed that for all n>=2, gpf(g(n))<=2.86sqrt(nlnn), and Grantham (1995) showed that for all n>=5, the constant 2.86 may be replaced by 1.328.

See also

Group Order, Symmetric Group

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Grantham, J. "The Largest Prime Dividing the Maximal Order of an Element of S_n." 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.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 S_n 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 g(n)." 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."

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Landau's Function

Cite this as:

Weisstein, Eric W. "Landau's Function." From MathWorld--A Wolfram Web Resource.

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