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A179010
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The number of isomorphism classes of commutative quandles of order n.
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2
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1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 7
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OFFSET
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1,9
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COMMENTS
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A quandle (X,*) is commutative if a*b = b*a for all a,b in X. Every finite commutative quandle (X,*) is obtained from an odd order, commutative Moufang loop (X,+) where x*y = (1/2)(x+y). Thus a(n) is the number of isomorphism classes of commutative Moufang loops of order n if n is odd and is 0 if n is even. Commutative Moufang loops of order less than 81 are associative hence abelian groups. But, there are two non-associative commutative Moufang loops of order 81. Thus a(n) = number of isomorphism classes of abelian groups of odd order for n < 81 and a(81) = A000688(81) + 2 = 7. For proofs of these facts see, e.g., the papers below by Belousov, Nagy and Vojtchovský, and Glauberman.
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LINKS
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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Results due to Belousov, Nagy and Vojtchovský, and Glauberman added, and sequence extended to n = 81, by W. Edwin Clark, Jan 25 2011
In Comments section, "Every commutative quandle" replaced with "Every finite commutative quandle" by W. Edwin Clark, Mar 09 2014
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STATUS
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approved
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