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A001423 Number of semigroups of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).
(Formerly M3550 N1438)
1, 1, 4, 18, 126, 1160, 15973, 836021, 1843120128, 52989400714478, 12418001077381302684 (list; graph; refs; listen; history; text; internal format)



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.

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

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

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


Table of n, a(n) for n=0..10.

A. de Vries, Formal Languages: An Introduction

Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).

Andreas Distler and Tom Kelsey, The Monoids of Order Eight and Nine, in Intelligent Computer Mathematics, Lecture Notes in Computer Science, Volume 5144/2008, Springer-Verlag. [From N. J. A. Sloane, Jul 10 2009]

A. Distler and T. Kelsey, The semigroups of order 9 and their automorphism groups, arXiv preprint arXiv:1301.6023 [math.CO], 2013.

Andreas Distler, Chris Jefferson, Tom Kelsey, Lars Kotthoff, The Semigroups of Order 10, in: M. Milano (Ed.), Principles and Practice of Constraint Programming, 18th International Conference, CP 2012, Qu├ębec City, QC, Canada, October 8-12, 2012, Proceedings (LNCS, volume 7514), pp. 883-899, Springer-Verlag Berlin Heidelberg 2012.

Remigiusz Durka, Kamil Grela, On the number of possible resonant algebras, arXiv:1911.12814 [hep-th], 2019.

G. E. Forsythe, SWAC computes 126 distinct semigroups of order 4, Proc. Amer. Math. Soc. 6, (1955). 443-447.

H. Juergensen and P. Wick, Die Halbgruppen von Ordnungen <= 7, Semigroup Forum, 14 (1977), 69-79.

H. Juergensen and P. Wick, Die Halbgruppen von Ordnungen <= 7, annotated and scanned copy.

Daniel J. Kleitman, Bruce L. Rothschild and Joel H. Spencer, The number of semigroups of order n, Proc. Amer. Math. Soc., 55 (1976), 227-232.

R. J. Plemmons, There are 15973 semigroups of order 6 (annotated and scanned copy)

Eric Postpischil Associativity Problem, Posting to sci.math newsgroup, May 21 1990.

S. Satoh, K. Yama, and M. Tokizawa, Semigroups of order 8, Semigroup Forum 49 (1994), 7-29.

N. J. A. Sloane, Overview of A001329, A001423-A001428, A258719, A258720.

T. Tamura, Some contributions of computation to semigroups and groupoids, pp. 229-261 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970. (Annotated and scanned copy)

Eric Weisstein's World of Mathematics, Semigroup.

Index entries for sequences related to semigroups


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


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

Sequence in context: A215691 A073511 A108704 * A308351 A291335 A158341

Adjacent sequences:  A001420 A001421 A001422 * A001424 A001425 A001426




N. J. A. Sloane


a(9) added by Andreas Distler, Jan 12 2011

a(10) from Distler et al, 2012, added by Andrey Zabolotskiy, Nov 08 2018



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Last modified January 16 12:02 EST 2021. Contains 340206 sequences. (Running on oeis4.)