A211168 Exponent of alternating group An.

1, 1, 3, 6, 30, 60, 420, 420, 1260, 2520, 27720, 27720, 360360, 360360, 360360, 360360, 6126120, 12252240, 232792560, 232792560, 232792560, 232792560, 5354228880, 5354228880, 26771144400, 26771144400, 80313433200, 80313433200, 2329089562800, 2329089562800

Offset: 1

Author

Alexander Gruber, Jan 31 2013

Keywords

• nonn
• nice

Comments

a(n) is the smallest natural number m such that g^m = 1 for any g in An.

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.

Examples

			For n = 7, lcm{1,...,5,7} = 420.
		

Links

• Alexander Gruber, Table of n, a(n) for n = 1..2308

Crossrefs

Even entries given by the sequence A076100, or the odd entries in the sequence A003418.

The records of this sequence are a subsequence of A002809 and A126098.

Programs

Magma

for n in [1..40] do
Exponent(AlternatingGroup(n));
end for;

Magma

for n in [1..40] do
if n mod 2 eq 0 then
L := [1..n-1];
else
L := Append([1..n-2],n);
end if;
LCM(L);
end for;

Mathematica

Table[If[Mod[n, 2] == 0, LCM @@ Range[n - 1],
  LCM @@ Join[Range[n - 2], {n}]], {n, 1, 100}] (* or *)
a[1] = 1; a[2] = 1; a[3] = 3; a[n_] := a[n] =
  If[Mod[n, 2] == 0, LCM[a[n - 1], n - 2], LCM[a[n - 2], n - 3, n]]; Table[a[n], {n, 1, 40}]

PARI

a(n)=lcm(if(n%2,concat([2..n-2],n),[2..n-1])) \\ Charles R Greathouse IV, Mar 02 2014

Formula

Explicit:

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

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

Recursive:

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

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

= lcm{a(n-2), n-3, n} if n is odd.

a(n) = A003418(n)/(1 + [n in A228693]) for n > 1. - Charlie Neder, Apr 25 2019