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A211168 Exponent of alternating group An. 1
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 (list; graph; refs; listen; history; text; internal format)
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 * A215294 A090932 A280981

Adjacent sequences:  A211165 A211166 A211167 * A211169 A211170 A211171

KEYWORD

nonn,nice

AUTHOR

Alexander Gruber, Jan 31 2013

STATUS

approved

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Last modified November 19 11:09 EST 2019. Contains 329319 sequences. (Running on oeis4.)