A001423 -id:A001423 - cp's OEIS frontend

A118581 Number of nonisomorphic semigroups of order <= n.

1, 2, 7, 31, 219, 2134, 30768, 1658440, 3685688857, 105981863625149

Offset: 0

Jonathan Vos Post, May 07 2006

Semigroup analog of A063756 Number of groups of order <= n. a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation (and thus is an associative groupoid). Some sources require that a semigroup have an identity element (in which case semigroups are identical to monoids). Not all sources agree that S should be nonempty. This sequence assumes that a semigroup may be empty and need not have an identity.

			a(7) = 1658440 = 1 + 1 + 5 + 24 + 188 + 1915 + 28634 + 1627672.

Cf. A001329, A001423, A001426, A023814, A027851, A029851, A058108, A063756, A079173.

a(n) = Sum_{i=0..n} A027851(i). a(n) = Sum_{i=0..n} (2*A001423(i) - A029851(i)).

a(8)-a(9) (using A027851) from Giovanni Resta, Jun 16 2016

A118601 Partial sums of A058129.

Original entry on oeis.org

1, 3, 10, 45, 273, 2510, 34069, 1703066

Offset: 1

Jonathan Vos Post, May 08 2006

Cf. A001329, A001423, A001426, A023814, A027851, A029851, A058108, A058132, A058133, A063756, A079173, A118581.

a(n) = SUM[i=1..n] A058129(i). a(n) = SUM[i=1..n] (2*A058133(i) - A058132(i)).

One more term from Jonathan Vos Post, Jul 20 2009

Edited by N. J. A. Sloane, Jul 25 2009

A001427 Number of regular semigroups of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

1, 3, 9, 42, 206, 1352, 10168, 91073, 925044

Offset: 1

N. J. A. Sloane

Tak-Shing T. Chan, YH Yang, Polar n-Complex and n-Bicomplex Singular Value Decomposition and Principal Component Pursuit, IEEE Transactions on Signal Processing ( Volume: 64, Issue: 24, Dec.15, 15 2016 ); DOI: 10.1109/TSP.2016.2612171
R. J. Plemmons, There are 15973 semigroups of order 6, Math. Algor., 2 (1967), 2-17; 3 (1968), 23.
R. J. Plemmons, Cayley Tables for All Semigroups of Order Less Than 7. Department of Mathematics, Auburn Univ., 1965.
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).

Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
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.
R. J. Plemmons, There are 15973 semigroups of order 6 (annotated and scanned copy)
S. Satoh, K. Yama, 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)
Index entries for sequences related to semigroups

a(8) and a(9) from Andreas Distler, Jan 17 2011

A118100 Number of commutative semigroups of order <= n.

Original entry on oeis.org

1, 2, 5, 17, 75, 400, 2543, 19834, 241639, 11787482, 3530717819

Offset: 0

Jonathan Vos Post, May 11 2006

A001426(n) is the number of commutative semigroups of order n. A001426(n) + A079193(n) + A079196(n) + A079199(n) = A001329(n). 2, 5, 17, 2543 and 241639 are primes.

			a(8) = 1 + 1 + 3 + 12 + 58 + 325 + 2143 + 17291 + 221805 = 241639.

Index entries for sequences related to semigroups.

Cf. A001423, A001426, A023815, A027851, A058105, A058116.

a(n) = Sum_{i=1..n} A001426(i).

a(9)-a(10) added using the terms in A001426 by Miles Englezou, May 27 2025

A186117 Number of nonisomorphic semigroups of order n minus number of groups of order n.

Original entry on oeis.org

0, 4, 23, 186, 1914, 28632, 1627671, 3684030412, 105978177936290

Offset: 1

Jonathan Vos Post, Feb 13 2011

In a sense, this measures the increase in combinatorial structures available by dropping the requirement of inverses, and an identity element, in moving from the group axioms to the semigroup axioms. A semigroup is mathematical object defined for a set and a binary operator in which the multiplication operation is associative. No other restrictions are placed on a semigroup; thus a semigroup need not have an identity element and its elements need not have inverses within the semigroup. Other sequences may be derived by considering commutative semigroups and commutative groups, self-converse semigroup, counting idempotents, and the like.

			a(1) = 0 because there are unique groups and semigroups of order 1, so 1 - 1  = 0.
a(2) = 4 because there are 5 semigroups of order 2 groups and a unique group of order 2, so 5 - 1  = 4.

Eric W. Weisstein, Finite Group
Eric W. Weisstein, Semigroup

Cf. A000001, A027851, A001423, A029851, A001426, A023814, A058108, A079173, A001329, A186116.

a(n) = A027851(n) - A000001(n).

A278565 a(n) = Sum_{t=1..n} binomial(n,t)*t^(1+(n-t)^2).

Original entry on oeis.org

0, 1, 4, 18, 236, 12760, 3162582, 5965957900, 147395915019656, 38431930179989653632, 90116582088416163834417290, 2118032070086776060232851050813004, 966490912699216393489571072350268614425420, 17165261065730992639912668446254005264689353839299152

Offset: 0

N. J. A. Sloane, Nov 25 2016

nonn

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. See Eq. (1.2).

Cf. A001423.

Table[Sum[Binomial[n, t]  t^(1 + (n - t)^2), {t, 1, n}], {n, 0, 25}] (* Vincenzo Librandi, Nov 27 2016 *)

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.

A383219 Number of nilpotent semigroups by order, up to isomorphism and anti-isomorphism.

Original entry on oeis.org

0, 0, 1, 2, 10, 93, 2813, 616830, 1833587417, 52972875977730

Offset: 0

Joseph E. Marrow, Apr 19 2025

Andreas Distler, Classification and Enumeration of Finite Semigroups, PhD Thesis, University of St. Andrews, (2010).

Cf. A001423.

cp's OEIS Frontend

A118581 Number of nonisomorphic semigroups of order <= n.

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A118601 Partial sums of A058129.

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A001427 Number of regular semigroups of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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A118100 Number of commutative semigroups of order <= n.

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A186117 Number of nonisomorphic semigroups of order n minus number of groups of order n.

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A278565 a(n) = Sum_{t=1..n} binomial(n,t)*t^(1+(n-t)^2).

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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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A383219 Number of nilpotent semigroups by order, up to isomorphism and anti-isomorphism.

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