A074859 Number of elements of S_n having the maximum possible order g(n), where g(n) is Landau's function (A000793).

1, 1, 1, 2, 6, 20, 240, 420, 2688, 18144, 120960, 2661120, 7983360, 103783680, 1037836800, 12454041600, 149448499200, 1693749657600, 60974987673600, 289631191449600, 5792623828992000, 121645100408832000, 3568256278659072000, 30776210403434496000, 738629049682427904000, 12310484161373798400000

Offset: 0

Author

Christopher J. Smyth, Sep 11 2002

Keywords

• easy
• nice
• nonn

References

• J.-L. Nicolas, On Landau's function g(n), pp. 228-240 of R. L. Graham et al., eds., Mathematics of Paul Erdős I.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..175
• J. Kuzmanovich and A. Pavlichenkov, Finite groups of matrices whose entries are integers, Amer. Math. Monthly, 109 (2002), 173-186.
• W. Miller, The Maximum Order of an Element of Finite Symmetric Group, Am. Math. Monthly, Jun-Jul 1987, pp. 497-506.
• J.-L. Nicolas, Sur l'ordre maximum d'un élément dans le groupe S_n des permutations, Acta Arith., 14 (1968), 315-332.
• J.-L. Nicolas, Ordre maximal d'un élément du groupe S_n de permutations et 'highly composite numbers', Bull. Math. Soc. France 97 (1969) 129-191.

Crossrefs

Cf. A000793 (Landau's function g(n)).

Cf. A074064, A074103, A074115, A162682.

Last row element of A057731. - Alois P. Heinz, Feb 14 2013

Programs

Mathematica

g[n_] := Max[ Apply[ LCM, IntegerPartitions[n], 1]]; f[x_, n_] := Total[ (MoebiusMu[g[n]/#]*Exp[ Total[ (x^#/# & ) /@ Divisors[#]]] & ) /@ Divisors[g[n]]]; a[n_] := n!*Coefficient[ Series[f[x, n], {x, 0, n}], x^n]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Nov 03 2011, after Vladeta Jovovic *)

Formula

a(n) = n!*coefficient of x^n in expansion of Sum_{i divides A000793(n)} mu(A000793(n)/i)*exp(Sum_{j divides i} x^j/j). - Vladeta Jovovic, Sep 29 2002

Extensions

Corrected and extended by Vladeta Jovovic, Sep 20 2002