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Largest order of even permutation of n elements, or maximal order of element of alternating group A_n.
8

%I #14 Jun 14 2024 22:31:10

%S 1,1,1,3,3,5,5,7,15,15,21,21,35,35,60,105,105,105,140,210,210,420,420,

%T 420,420,840,1155,1260,1365,1540,2310,2520,4620,4620,5460,5460,9240,

%U 9240,13860,15015,16380,16380,27720,30030,32760,60060,60060,60060

%N Largest order of even permutation of n elements, or maximal order of element of alternating group A_n.

%D J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].

%D V. Jovovic, Some combinatorial characteristics of symmetric and alternating groups (in Russian), Belgrade, 1980, unpublished.

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

%F a(n)=max{ A000793(n-2), A051704(n-1), A051704(n) }, a(0)=a(1)=1.

%t (* a3 = A000793 a4 = A051704 *) a3[n_] := Max[LCM @@@ IntegerPartitions[n]]; a4[n_] := (pp = Reap[ Do[ pk = p^k; If[pk <= n, Sow[pk]], {p, Prime[ Range[2, PrimePi[n]]]}, {k, 1, Ceiling[ Log[3, n]]}]][[2, 1]]; sel = Select[ IntegerPartitions[n, All, pp], Length[#] == Length[ Union[#] && !MatchQ[#, {___, x_, ___, y_, ___} /; GCD[x, y] != 1]] &]; Max[Times @@@ sel]); a4[0] = 1; a4[1] = a4[2] = a4[4] = a4[6] = 0; a[n_] := Max[a3[n - 2], a4[n - 1], a4[n]]; a[0] = a[1] = a[2] = 1; Table[a[n], {n, 0, 47}] (* _Jean-François Alcover_, Sep 11 2012, from formula *)

%o (PARI) a(n)={my(m=1); forpart(p=n, if(sum(i=1, #p, p[i]-1)%2==0, m=max(m, lcm(Vec(p))))); m} \\ _Andrew Howroyd_, Jul 03 2018

%Y Cf. A057742, A057743, A057740, A000793.

%K nonn,nice,easy

%O 0,4

%A _Vladeta Jovovic_