A057996 -id:A057996 - cp's OEIS frontend

A057771 Number of loops (quasigroups with an identity element) of order n.

0, 1, 1, 1, 2, 6, 109, 23746, 106228849, 9365022303540, 20890436195945769617, 1478157455158044452849321016

Offset: 0

Christian G. Bower, Nov 01 2000

Brendan D. McKay, A. Meynert and W. Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Designs 15 (2007), no. 2, 98-119.
A. Hulpke, Petteri Kaski and Patric R. J. Östergård, The number of Latin squares of order 11, Math. Comp. 80 (2011) 1197-1219.
Index entries for sequences related to quasigroups

Cf. A000315, A057991, A057992, A057993, A057994, A057996, A057997, A057998, A089925.

a(8) from Juergen Buntrock (jubu(AT)jubu.com), Nov 03 2003.
a(9)-a(10) (from the McKay-Meynert-Myrvold article) from Richard Bean, Feb 17 2004
a(11) from Petteri Kaski (petteri.kaski(AT)cs.helsinki.fi), Sep 18 2009
a(0) prepended by Jianing Song, Oct 26 2019

A057991 Number of quasigroups of order n.

Original entry on oeis.org

1, 1, 1, 5, 35, 1411, 1130531, 12198455835, 2697818331680661, 15224734061438247321497, 2750892211809150446995735533513, 19464657391668924966791023043937578299025

Offset: 0

Christian G. Bower, Nov 01 2000

A. Hulpke, Petteri Kaski and Patric R. J. Östergård, The number of Latin squares of order 11, Math. Comp. 80 (2011) 1197-1219.
Brendan D. McKay, A. Meynert and W. Myrvold, Small Latin Squares, Quasigroups and Loops, J. Combin. Designs 15 (2007), no. 2, 98-119.
D. S. Stones, The parity of the number of quasigroups, Discr. Math., 310 (2010), 3033-3039. [From _N. J. A. Sloane_, Sep 25 2010]
Index entries for sequences related to quasigroups

Cf. A002860, A057992, A057993, A057994, A057771, A057996.

More terms (from the McKay-Meynert-Myrvold article) from Richard Bean, Feb 17 2004

a(11) from Petteri Kaski (petteri.kaski(AT)cs.helsinki.fi), Sep 18 2009

A057994 Number of asymmetric quasigroups of order n.

Original entry on oeis.org

1, 1, 1, 0, 16, 1303, 1127628

Offset: 0

Christian G. Bower, Nov 01 2000

Index entries for sequences related to quasigroups

Cf. A057991-A057993, A057771, A057996.

A057992 Number of commutative quasigroups of order n.

Original entry on oeis.org

1, 1, 1, 3, 7, 11, 491, 6381, 10940111, 1225586965, 130025302505741, 252282619993126717, 2209617218725712597768722, 98758655816833782283724345637

Offset: 0

Christian G. Bower, Nov 01 2000

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130.
Index entries for sequences related to quasigroups

Cf. A057991-A057994, A057771, A057996, A035481, A350009, A350010.

Added a(7) = 6381, W. Edwin Clark, Jan 04 2011

a(8)-a(13) from Ian Wanless, Dec 08 2021

A057993 Number of self-converse quasigroups of order n.

Original entry on oeis.org

1, 1, 1, 3, 13, 81, 3883

Offset: 0

Christian G. Bower, Nov 01 2000

Index entries for sequences related to quasigroups

Cf. A057991-A057994, A057771, A057996.

A057998 Number of asymmetric loops (quasigroups with an identity element).

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 61, 23374, 106185390

Offset: 0

Christian G. Bower, Nov 20 2000

nonn

Index entries for sequences related to quasigroups

Cf. A057996.

a(8) from Juergen Buntrock (jubu(AT)eule.in-berlin.de), Nov 17 2003

A118127 Number of quasigroups of order <= n.

Original entry on oeis.org

1, 2, 3, 8, 43, 1454, 1131985, 12199587820, 2697830531268481, 15224736759268778589978, 2750892227033887206264514123491

Offset: 1

Jonathan Vos Post, May 12 2006

nonn

A quasigroup is a groupoid G such that for all a and b in G, there exist unique c and d in G such that ac = b and da = b. Hence a quasigroup is not required to have an identity element, nor be associative. Equivalently, one can state that quasigroups are precisely groupoids whose multiplication tables are Latin squares (possibly empty).

			a(10) = 2750892227033887206264514123491 = 1 + 1 + 1 + 5 + 35 + 1411 + 1130531 + 12198455835 + 2697818331680661 + 15224734061438247321497 + 2750892211809150446995735533513.

Index entries for sequences related to quasigroups.

Cf. A002860, A057991-A057994, A057771, A057996, A118641.

a(n) = SUM[i=0..n] A057991(i).

A328746 Number of loops of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

0, 1, 1, 1, 2, 5, 72, 12151, 53146457

Offset: 0

Jianing Song, Oct 26 2019

For the number of group-like algebraic structures of order n, see:
Semigroups: A027851 or A001423 (commutative: A001426);
Monoids: A058129 or A058133 (commutative: A058131);
Quasigroups: A057991 or A058171 (commutative: A057992);
Loops: A057771 or this sequence (commutative: A089925);
Groups: A000001 (commutative: A000688);
Rings: A027623 or A038036 (commutative: A037289);
Rings with unity: A037291;
Fields: A069513.

a(n) = (A057771(n)+A057996(n))/2.

cp's OEIS Frontend

A057771 Number of loops (quasigroups with an identity element) of order n.

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A057991 Number of quasigroups of order n.

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A057994 Number of asymmetric quasigroups of order n.

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A057992 Number of commutative quasigroups of order n.

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A057993 Number of self-converse quasigroups of order n.

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A057998 Number of asymmetric loops (quasigroups with an identity element).

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A118127 Number of quasigroups of order <= n.

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A328746 Number of loops of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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