%I
%S 1,1,3,6,30,60,420,420,1260,2520,27720,27720,360360,360360,360360,
%T 360360,6126120,12252240,232792560,232792560,232792560,232792560,
%U 5354228880,5354228880,26771144400,26771144400,80313433200,80313433200,2329089562800,2329089562800
%N Exponent of alternating group An.
%C a(n) is the smallest natural number m such that g^m = 1 for any g in An.
%C If m <= n, a mcycle occurs in some permutation in An if and only if m is odd or m <= n  2. The exponent is the LCM of the m's satisfying these conditions, leading to the formula below.
%H Alexander Gruber, <a href="/A211168/b211168.txt">Table of n, a(n) for n = 1..2308</a>
%F Explicit:
%F a(n) = lcm{1,..., n1} if n is even.
%F = lcm{1,..., n2, n} if n is odd.
%F Recursive:
%F Let a(1) = a(2) = 1 and a(3) = 3. Then
%F a(n) = lcm{a(n1), n2} if n is even.
%F = lcm{a(n2), n3, n} if n is odd.
%F a(n) = A003418(n)/(1 + [n in A228693]) for n > 1.  _Charlie Neder_, Apr 25 2019
%e For n = 7, lcm{1,...,5,7} = 420.
%t Table[If[Mod[n, 2] == 0, LCM @@ Range[n  1],
%t LCM @@ Join[Range[n  2], {n}]], {n, 1, 100}] (* or *)
%t a[1] = 1; a[2] = 1; a[3] = 3; a[n_] := a[n] =
%t If[Mod[n, 2] == 0, LCM[a[n  1], n  2], LCM[a[n  2], n  3, n]]; Table[a[n], {n, 1, 40}]
%o (MAGMA)
%o for n in [1..40] do
%o Exponent(AlternatingGroup(n));
%o end for;
%o (MAGMA)
%o for n in [1..40] do
%o if n mod 2 eq 0 then
%o L := [1..n1];
%o else
%o L := Append([1..n2],n);
%o end if;
%o LCM(L);
%o end for;
%o (PARI) a(n)=lcm(if(n%2,concat([2..n2],n),[2..n1])) \\ _Charles R Greathouse IV_, Mar 02 2014
%Y Even entries given by the sequence A076100, or the odd entries in the sequence A003418.
%Y The records of this sequence are a subsequence of A002809 and A126098.
%K nonn,nice
%O 1,3
%A _Alexander Gruber_, Jan 31 2013
