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A211168
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Exponent of alternating group An.
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1
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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For n = 7, lcm{1,...,5,7} = 420.
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MATHEMATICA
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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}]
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PROG
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(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;
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CROSSREFS
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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.
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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