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A193067 The number of isomorphism classes of connected Alexander (aka indecomposable affine) quandles of order n 2
1, 0, 1, 1, 3, 0, 5, 2, 8, 0, 9, 1, 11, 0, 3, 9, 15, 0, 17, 3, 5, 0, 21, 2, 34, 0, 30, 5, 27, 0, 29, 8, 9, 0, 15, 8, 35, 0, 11, 6, 39, 0, 41, 9, 24, 0, 45, 9, 76, 0, 15, 11, 51, 0, 27, 10, 17, 0, 57, 3, 59, 0, 40, 61, 33, 0, 65, 15, 21, 0, 69, 16, 71, 0, 34 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Finite connected Alexander (affine) quandles are Latin. According to the Toyoda-Bruck theorem, Latin affine quandles are the same objects as idempotent medial quasigroups. The values up to 16 were obtained by Nelson (see links below). - Edited by David Stanovsky, Oct 01 2014

LINKS

W. Edwin Clark, Table of n, a(n) for n = 1..255

W. E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, Quandle Colorings of Knots and Applications, arXiv preprint arXiv:1312.3307, 2013

S. Nelson, Classification of Finite Alexander Quandles

S. Nelson, Alexander Quandles of Order 16

K. Toyoda, On axioms of linear functions/a>, Proceedings of the Imperial Academy 17/7(1941), 221-227.

Wikipedia, Medial

PROG

(GAP)

findY:=function(f, g)

local Y, y;

  Y:=[];

  for y in g do

    Add(Y, Image(f, y^(-1))*y);

  od;

  Y:=Set(Y);

  return Subgroup(g, Y);

end;;

CA:=[];;

k:=8;;

for n in [1..2^k-1] do

   CA[n]:=0;

   LGn:=AllSmallGroups(n, IsAbelian);

   for g in LGn do

     autg:=AutomorphismGroup(g);;

     eautg:=List(ConjugacyClasses(autg), Representative);

     for f in eautg do

       N2:=findY(f, g);

       if Size(N2) = n then CA[n]:=CA[n]+1; fi;

     od;

   od;

   for j in [1..k] do

   if n = 2^j and n <> 2^(j-1) then Print("done to ", n, "\n"); fi;

   od;

od;

for n in [1..2^k-1] do

  Print(CA[n], ", ");

od;

CROSSREFS

See Index to OEIS under quandles.

Sequence in context: A098496 A175297 A165754 * A177886 A011293 A181884

Adjacent sequences:  A193064 A193065 A193066 * A193068 A193069 A193070

KEYWORD

nonn

AUTHOR

W. Edwin Clark, Jul 15 2011

STATUS

approved

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Last modified February 19 22:04 EST 2018. Contains 299357 sequences. (Running on oeis4.)