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A196111
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Number of isomorphism classes of simple quandles of order n.
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1
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1, 1, 1, 3, 0, 5, 2, 3, 1, 9, 1, 11, 0, 2, 3, 15, 0, 17, 2, 2, 0, 21, 1, 10, 0, 8, 2, 27, 1, 29, 6, 0, 0, 0, 3, 35, 0, 0, 2, 39, 3, 41, 0, 3, 0, 45
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OFFSET
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2,4
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COMMENTS
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A quandle is simple if it has more than one element, and if it has no homomorphic images other than itself or the singleton quandle. Since a simple quandle with more than two elements is connected, we have a(n) <= A181771(n), for n > 2, with equality if n is prime.
Some authors consider the quandle with one element to be simple and some do not.
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LINKS
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Table of n, a(n) for n=2..47.
W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, Quandle Colorings of Knots and Applications, arXiv preprint arXiv:1312.3307, 2013
David Joyce, Simple Quandles, J. Algebra 79(2) 1982, 307-318.
Leandro Vendramin, On the classification of quandles of low order, arXiv:1105.5341v1 [math.GT].
Leandro Vendramin and Matías Graña, Rig, a GAP package for racks and quandles.
Wikipedia, Racks and quandles
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FORMULA
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a(p) = A181771(p) = p - 2, for prime p > 2.
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EXAMPLE
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a(2) = 1 since the quandle of order 2 is trivially simple (though not connected).
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PROG
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(GAP) (using the Rig package)
LoadPackage("rig");
IsSimpleQuandle:=function(q)
local g, N, gg, n;
if IsFaithful(q) = false then return false; fi;
g:=InnerGroup(q);;
if Size(Center(g))>1 then return false; fi;
N:=NormalSubgroups(g);;
gg:=DerivedSubgroup(g);;
for n in N do
if Size(n) = 1 then continue; fi;
if IsSubset(gg, n) and Size(n)<Size(gg) then return false; fi;
od;
return true;
end;;
a:=[1, 1];;
for n in [3..35] do
a[n]:=0;
for i in [1..NrSmallQuandles(n)] do
if IsSimpleQuandle(SmallQuandle(n, i)) then
a[n]:=a[n]+1;
fi;
od;
od;
List([1..35], u->a[u]); # W. Edwin Clark, Dec 06 2011
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CROSSREFS
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Cf. A181769, A181771.
See also Index to OEIS under quandles.
Sequence in context: A324103 A130054 A236146 * A261628 A007431 A215447
Adjacent sequences: A196108 A196109 A196110 * A196112 A196113 A196114
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KEYWORD
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nonn,hard,more
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AUTHOR
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James McCarron, Oct 27 2011
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EXTENSIONS
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a(21) corrected by W. Edwin Clark, Dec 06 2011
a(32)-a(35) added by W. Edwin Clark, Dec 06 2011
a(36)-a(47) added by W. Edwin Clark, Dec 28 2014
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STATUS
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approved
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