A001329 -id:A001329 - cp's OEIS frontend

A002489 a(n) = n^(n^2), or (n^n)^n.

1, 1, 16, 19683, 4294967296, 298023223876953125, 10314424798490535546171949056, 256923577521058878088611477224235621321607, 6277101735386680763835789423207666416102355444464034512896, 196627050475552913618075908526912116283103450944214766927315415537966391196809

Offset: 0

N. J. A. Sloane

The number of closed binary operations on a set of order n. Labeled groupoids.
The values of "googol" in base N: "10^100" in base 2 is 2^4=16; "10^100" in base 3 is 3^9=19683, etc. This is N^^3 by the "lower-valued" (left-associative) definition of the hyper4 or tetration operator (see Munafo webpage). - Robert Munafo, Jan 25 2010
n^(n^k) = (((n^n)^n)^...)^n, with k+1 n's, k >= 0. - Daniel Forgues, May 18 2013

			a(3) = 19683 because (3^3)^3 = 3^(3^2) = 19683.

John S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 6.
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).

Michael Lee, Table of n, a(n) for n = 0..26 (first 16 terms from Vincenzo Librandi)
Robert Munafo, Hyper4 Iterated Exponential Function [From _Robert Munafo_, Jan 25 2010]
Eric Postpischil, Posting to sci.math newsgroup, May 21 1990.
P. Rossier, Grands nombres, Elemente der Mathematik, Vol. 3 (1948), p. 20; alternative link.
Index entries for sequences related to groupoids

a(n) = A079172(n) + A023814(n) = A079176(n) + A079179(n);
a(n) = A079182(n) + A023813(n) = A079186(n) + A079189(n);
a(n) = A079192(n) + A079195(n) + A079198(n) + A023815(n).
Cf. A002488, A001329, A002488, A023813, A076113, A090588, A000312, A258102.

[n^(n^2): n in [0..10]]; // Vincenzo Librandi, May 13 2011

Join[{1},Table[n^n^2,{n,10}]] (* Harvey P. Dale, Sep 06 2011 *)

a(n)=n^(n^2) \\ Charles R Greathouse IV, Nov 20 2012

a(n) = [x^(n^2)] 1/(1 - n*x). - Ilya Gutkovskiy, Oct 10 2017

Sum_{n>=1} 1/a(n) = A258102. - Amiram Eldar, Nov 11 2020

A001423 Number of 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, 1, 4, 18, 126, 1160, 15973, 836021, 1843120128, 52989400714478, 12418001077381302684

Offset: 0

N. J. A. Sloane

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).

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, and Lars Kotthoff, The Semigroups of Order 10, 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.
Remigiusz Durka and 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

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

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

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

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

A027851 Number of nonisomorphic semigroups of order n.

Original entry on oeis.org

1, 1, 5, 24, 188, 1915, 28634, 1627672, 3684030417, 105978177936292

Offset: 0

Christian G. Bower, Dec 13 1997, updated Feb 19 2001

Özgur Akgun, Mun See Chang, Ian P. Gent, and Christopher Jefferson, Breaking the Symmetries of Indistinguishable Objects, arXiv:2503.17251 [cs.AI], 2025. See pp. 14, 17.
Peter Cameron's Blog, The combinatorial explosion, Posted 18/02/2016.
Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
A. Distler and T. Kelsey, The semigroups of order 9 and their automorphism groups, arXiv preprint arXiv:1301.6023 [math.CO], 2013.
C. Noebauer, Home page
C. Noebauer, The Numbers of Small Rings
C. Noebauer, Thesis on the enumeration of near-rings
Eric Postpischil Posting to sci.math newsgroup, May 21 1990
Arman Shamsgovara, Enumerating, Cataloguing and Classifying All Quantales on up to Nine Elements, In: Glück, R., Santocanale, L., and Winter, M. (eds), Relational and Algebraic Methods in Computer Science (RAMiCS 2023) Lecture Notes in Computer Science, Springer, Cham, Vol. 13896.
Jeremy G. Sumner, Michael D. Woodhams, Lie-Markov models derived from finite semigroups, arXiv:1709.00520 [math.GR], 2017.
Michael Torpey, Semigroup congruences: computational techniques and theoretical applications, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).
A. de Vries, Formal Languages: An Introduction, 2012.
Eric Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

Cf. A001426, A023814, A058108.

Cf. A001423, A029851, A079173, A001329.

a(n) = A001423(n)*2 - A029851(n).

a(n) + A079173(n) = A001329(n).

a(8)-a(9) from Andreas Distler, Jan 13 2011

A001426 Number of commutative semigroups of order n.

Original entry on oeis.org

1, 1, 3, 12, 58, 325, 2143, 17291, 221805, 11545843, 3518930337

Offset: 0

N. J. A. Sloane

P. A. Grillet, Computing Finite Commutative Semigroups, Semigroup Forum 53 (1996), 140-154.
P. A. Grillet, Computing Finite Commutative Semigroups: Part II, Semigroup Forum 67 (2003), 159-184.
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).

Remigiusz Durka, Kamil Grela, On the number of possible resonant algebras, arXiv:1911.12814 [hep-th], 2019.
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)
Eric Postpischil Posting to sci.math newsgroup, May 21 1990 [Broken link]
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)
Eric Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

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

a(n) + A079193(n) + A079196(n) + A079199(n) = A001329(n).

a(8) (from the Satoh et al. paper) supplied by Richard C. Schroeppel, Jul 22 2005

a(9) and a(10) from Grillet references sent by Jens Zumbragel (jzumbr(AT)math.unizh.ch), Jun 14 2006

A079171 Number of isomorphism classes of closed binary operations (groupoids) on a set of order n, listed by class size.

Original entry on oeis.org

1, 4, 6, 3, 12, 78, 3237, 2, 1, 14, 30, 275, 495, 48810, 178932325

Offset: 1

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
A002489(n) is equal to the sum of the products of each element in row n of this sequence and the corresponding element of A079210.
The sum of each row n is given by A001329(n).

			First four rows:
  1;
  4, 6;
  3, 12, 78, 3237;
  2, 1, 14, 30, 275, 495, 48810, 178932325.

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A002489, A001329. a(n, A027423(n)) = A030245(n).

A001425 Number of commutative groupoids with n elements.

Original entry on oeis.org

1, 1, 4, 129, 43968, 254429900, 30468670170912, 91267244789189735259, 8048575431238519331999571800, 24051927835861852500932966021650993560, 2755731922430783367615449408031031255131879354330

Offset: 0

N. J. A. Sloane

nonn

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

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

a(n)+A079183(n)=A001329(n)

Cf. A001329, A023813, A038016.

a(n) = sum {1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s2!*...)) where fixA[s_1, s_2, ...] = prod {i>=j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j, odd} (sum {d|i} (d*s_d))^((i*s_i^2+s_i)/2) or {i=j, even} (sum {d|i} (d*s_d))^(i*s_i^2/2) * (sum {d|i/2} (d*s_d))^s_i or {i != j} (sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j)

a(n) asymptotic to (n^binomial(n+1, 2))/n! = A023813(n)/A000142(n) ~ e^n*n^binomial(n, 2) / sqrt(2*pi*n).

More terms from Christian G. Bower Feb 15 1998 and May 15 1998. Formula Dec 03 2003.

A079196 Number of isomorphism classes of non-associative commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 1, 117, 43910, 254429575, 30468670168769, 91267244789189717968, 8048575431238519331999349995, 24051927835861852500932966021639447717, 2755731922430783367615449408031031255128360423993

Offset: 0

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

nonn

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Row sums of A079197.

Cf. A001329, A001425, A001426, A079193, A079195 (labeled case), A079199.

A079193(n) + a(n) + A079199(n) + A001426(n) = A001329(n).

a(n) = A001425(n) - A001426(n). - Andrew Howroyd, Jan 26 2022

a(0)=0 prepended and a(5)-a(10) added by Andrew Howroyd, Jan 26 2022

A001428 Number of inverse 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, 2, 5, 16, 52, 208, 911, 4637, 26422, 169163, 1198651, 9324047, 78860687, 719606005, 7035514642

Offset: 1

N. J. A. Sloane

S. Satoh, K. Yama, and M. Tokizawa, Semigroups of order 8, Semigroup Forum 49 (1994), 7-29.
H. Juergensen and P. Wick, Die Halbgruppen von Ordnungen <= 7, Semigroup Forum, 14 (1977), 69-79.
R. J. Plemmons, Cayley Tables for All Semigroups of Order Less Than 7. Department of Mathematics, Auburn Univ., 1965.
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).
M. V. Lawson, Inverse Semigroups: The Theory of Partial Symmetries, World Scientific, 1998. [From Jonathan Vos Post, Mar 08 2010]
G. B. Preston, "Inverse semi-groups". Journal of the London Mathematical Society 29: 396-403. [From Jonathan Vos Post, Mar 08 2010]
V. V. Wagner (1952). "Generalised groups". Proceedings of the USSR Academy of Sciences 84: 1119-1122. (Russian) English translation. [From Jonathan Vos Post, Mar 08 2010]

Joao Araujo and Michael Kinyon, An elegant 3-basis for inverse semigroups, March 21, 2010. [From _Jonathan Vos Post_, Mar 23 2010]
Andreas Distler, Classification and Enumeration of Finite Semigroups, A Thesis Submitted for the Degree of PhD, University of St Andrews (2010).
Luna Elliott, Alex Levine, and James Mitchell, E-disjunctive inverse semigroups, arXiv:2405.19825 [math.GR], 2024. See p. 3.
H. Juergensen and P. Wick, Die Halbgruppen von Ordnungen <= 7, annotated and scanned copy.
Martin E. Malandro, Enumeration of finite inverse semigroups, arXiv:1312.7192 [math.CO]
R. J. Plemmons, There are 15973 semigroups of order 6 (annotated and scanned copy)
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)
Wikipedia, Inverse semigroup
Index entries for sequences related to semigroups

Cf. A234843 (commutative inverse semigroups), A234844 (inverse monoids), A234845 (commutative inverse monoids).

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

Added more terms (from the Malandro reference), Joerg Arndt, Dec 30 2013

A079177 Number of isomorphism classes of non-anti-associative closed binary operations on a set of order n.

Original entry on oeis.org

0, 8, 3320, 178964172

Offset: 1

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

nonn

A079177(n)+A079180(n)=A001329(n).

Each a(n) is equal to the sum of the elements in row n of A079178.

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A079176, A079178, A079180.

A079199 Number of isomorphism classes of associative non-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 2, 12, 130, 1590, 26491, 1610381, 3683808612, 105978166390449

Offset: 1

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

nonn

A079193(n)+A079196(n)+A079199(n)+A001426(n)=A001329(n).
Each a(n) is equal to the sum of the elements in row n of A079200.
Since this is the number of nonisomorphic noncommutative semigroups of order n, A079199(n)=A027851(n)-A001426(n). - Stanislav Sykora, Apr 03 2016

C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to semigroups

Cf. A001426, A027851, A079193, A079196, A079198, A079200.

Added terms a(5)-a(9). - Stanislav Sykora, Apr 03 2016

cp's OEIS Frontend

A002489 a(n) = n^(n^2), or (n^n)^n.

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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).

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A027851 Number of nonisomorphic semigroups of order n.

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A001426 Number of commutative semigroups of order n.

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A079171 Number of isomorphism classes of closed binary operations (groupoids) on a set of order n, listed by class size.

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A001425 Number of commutative groupoids with n elements.

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A079196 Number of isomorphism classes of non-associative commutative closed binary operations on a set of order n.

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A001428 Number of inverse semigroups of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

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Extensions

A079177 Number of isomorphism classes of non-anti-associative closed binary operations on a set of order n.

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