%I M3550 N1438 #71 Mar 24 2021 11:32:59

%S 1,1,4,18,126,1160,15973,836021,1843120128,52989400714478,

%T 12418001077381302684

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

%D David Nacin, "Puzzles, Parity Maps, and Plenty of Solutions", Chapter 15, The Mathematics of Various Entertaining Subjects: Volume 3 (2019), Jennifer Beineke & Jason Rosenhouse, eds. Princeton University Press, Princeton and Oxford, p. 245.

%D R. J. Plemmons, There are 15973 semigroups of order 6, Math. Algor., 2 (1967), 2-17; 3 (1968), 23.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H A. de Vries, <a href="http://haegar.fh-swf.de/Seminare/Genome/Archiv/languages.pdf">Formal Languages: An Introduction</a>

%H Andreas Distler, <a href="http://hdl.handle.net/10023/945">Classification and Enumeration of Finite Semigroups</a>, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).

%H Andreas Distler and Tom Kelsey, <a href="http://hpc.gap-system.org/Preprints/DK-AISC08-cr2.pdf">The Monoids of Order Eight and Nine</a>, in Intelligent Computer Mathematics, Lecture Notes in Computer Science, Volume 5144/2008, Springer-Verlag. [From _N. J. A. Sloane_, Jul 10 2009]

%H A. Distler and T. Kelsey, <a href="http://arxiv.org/abs/1301.6023">The semigroups of order 9 and their automorphism groups</a>, arXiv preprint arXiv:1301.6023 [math.CO], 2013.

%H Andreas Distler, Chris Jefferson, Tom Kelsey, and Lars Kotthoff, <a href="https://doi.org/10.1007/978-3-642-33558-7_63">The Semigroups of Order 10</a>, in: M. Milano (Ed.), Principles and Practice of Constraint Programming, 18th International Conference, CP 2012, Québec City, QC, Canada, October 8-12, 2012, Proceedings (LNCS, volume 7514), pp. 883-899, Springer-Verlag Berlin Heidelberg 2012.

%H Remigiusz Durka and Kamil Grela, <a href="https://arxiv.org/abs/1911.12814">On the number of possible resonant algebras</a>, arXiv:1911.12814 [hep-th], 2019.

%H G. E. Forsythe, <a href="https://doi.org/10.1090/S0002-9939-1955-0069814-7">SWAC computes 126 distinct semigroups of order 4</a>, Proc. Amer. Math. Soc. 6, (1955). 443-447.

%H H. Juergensen and P. Wick, <a href="http://dx.doi.org/10.1007/BF02194655">Die Halbgruppen von Ordnungen <= 7</a>, Semigroup Forum, 14 (1977), 69-79.

%H H. Juergensen and P. Wick, <a href="/A001423/a001423.pdf">Die Halbgruppen von Ordnungen <= 7</a>, annotated and scanned copy.

%H Daniel J. Kleitman, Bruce L. Rothschild and Joel H. Spencer, <a href="https://doi.org/10.1090/S0002-9939-1976-0414380-0">The number of semigroups of order n</a>, Proc. Amer. Math. Soc., 55 (1976), 227-232.

%H R. J. Plemmons, <a href="/A001423/a001423_2.pdf">There are 15973 semigroups of order 6</a> (annotated and scanned copy)

%H Eric Postpischil <a href="https://groups.google.com/forum/?hl=en#!msg/sci.math/nU-hg-FFSFo/iIB3Lul1sAEJ">Associativity Problem</a>, Posting to sci.math newsgroup, May 21 1990.

%H S. Satoh, K. Yama, and M. Tokizawa, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN001263501">Semigroups of order 8</a>, Semigroup Forum 49 (1994), 7-29.

%H N. J. A. Sloane, <a href="/A001329/a001329.jpg">Overview of A001329, A001423-A001428, A258719, A258720.</a>

%H T. Tamura, <a href="/A001329/a001329.pdf">Some contributions of computation to semigroups and groupoids</a>, pp. 229-261 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970. (Annotated and scanned copy)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Semigroup.html">Semigroup.</a>

%H <a href="/index/Se#semigroups">Index entries for sequences related to semigroups</a>

%F a(n) = (A027851(n) + A029851(n))/2.

%Y Cf. A001426, A023814, A058107, A058123, A151823.

%K nonn,hard,more,nice

%O 0,3

%A _N. J. A. Sloane_

%E a(9) added by _Andreas Distler_, Jan 12 2011

%E a(10) from Distler et al. 2012, added by _Andrey Zabolotskiy_, Nov 08 2018