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Number of elements of S_n having the maximum possible order g(n), where g(n) is Landau's function (A000793).
10

%I #32 Aug 16 2017 18:13:46

%S 1,1,1,2,6,20,240,420,2688,18144,120960,2661120,7983360,103783680,

%T 1037836800,12454041600,149448499200,1693749657600,60974987673600,

%U 289631191449600,5792623828992000,121645100408832000,3568256278659072000,30776210403434496000,738629049682427904000,12310484161373798400000

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

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

%H Alois P. Heinz, <a href="/A074859/b074859.txt">Table of n, a(n) for n = 0..175</a>

%H J. Kuzmanovich and A. Pavlichenkov, <a href="http://www.jstor.org/stable/2695329">Finite groups of matrices whose entries are integers</a>, Amer. Math. Monthly, 109 (2002), 173-186.

%H W. Miller, <a href="http://www.jstor.org/stable/2322839">The Maximum Order of an Element of Finite Symmetric Group</a>, Am. Math. Monthly, Jun-Jul 1987, pp. 497-506.

%H J.-L. Nicolas, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa14/aa14120.pdf">Sur l'ordre maximum d'un élément dans le groupe S_n des permutations</a>, Acta Arith., 14 (1968), 315-332.

%H J.-L. Nicolas, <a href="http://archive.numdam.org/article/BSMF_1969__97__129_0.pdf">Ordre maximal d'un élément du groupe S_n de permutations et 'highly composite numbers'</a>, Bull. Math. Soc. France 97 (1969) 129-191.

%F 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

%t 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_ *)

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

%Y Cf. A074064, A074103, A074115, A162682.

%Y Last row element of A057731. - _Alois P. Heinz_, Feb 14 2013

%K easy,nice,nonn

%O 0,4

%A _Christopher J. Smyth_, Sep 11 2002

%E Corrected and extended by _Vladeta Jovovic_, Sep 20 2002