A051593 Largest order of even permutation of n elements, or maximal order of element of alternating group A_n.

1, 1, 1, 3, 3, 5, 5, 7, 15, 15, 21, 21, 35, 35, 60, 105, 105, 105, 140, 210, 210, 420, 420, 420, 420, 840, 1155, 1260, 1365, 1540, 2310, 2520, 4620, 4620, 5460, 5460, 9240, 9240, 13860, 15015, 16380, 16380, 27720, 30030, 32760, 60060, 60060, 60060

Offset: 0

Author

Vladeta Jovovic

Keywords

• nonn
• nice
• easy

References

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• V. Jovovic, Some combinatorial characteristics of symmetric and alternating groups (in Russian), Belgrade, 1980, unpublished.

Links

• Index entries for sequences related to groups

Crossrefs

Cf. A057742, A057743, A057740, A000793.

Programs

Mathematica

(* 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 *)

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

Formula

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