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

1,3

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.

Links

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

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

Example

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

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}]

Prog

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

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.

Sequence in context: A136944 A136946 A125521 * A355989 A215294 A350756

Adjacent sequences: A211165 A211166 A211167 * A211169 A211170 A211171

Keyword

nonn,nice

Author

Alexander Gruber, Jan 31 2013

Status

approved