cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A211168 Exponent of alternating group An.

Original entry on oeis.org

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

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Author

Alexander Gruber, Jan 31 2013

Keywords

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.
		

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