A211168 - OEIS

%I #51 Feb 16 2022 23:40:56

%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 m-cycle 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, ..., n-1} if n is even.

%F      = lcm{1, ..., n-2, n} if n is odd.

%F Recursive:

%F Let a(1) = a(2) = 1 and a(3) = 3.  Then

%F a(n) = lcm{a(n-1), n-2} if n is even.

%F      = lcm{a(n-2), n-3, 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..n-1];

%o else

%o L := Append([1..n-2],n);

%o end if;

%o LCM(L);

%o end for;

%o (PARI) a(n)=lcm(if(n%2,concat([2..n-2],n),[2..n-1])) \\ _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