A023815 -id:A023815 - 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

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

A023814 Number of associative binary operations on an n-set; number of labeled semigroups.

Original entry on oeis.org

1, 1, 8, 113, 3492, 183732, 17061118, 7743056064, 148195347518186, 38447365355811944462

Offset: 0

Lyle Ramshaw (ramshaw(AT)pa.dec.com)

Alex Bailey, Martin Finn-Sell, and Robert Snocken, Subsemigroup, ideal and congruence growth of free semigroups, arXiv preprint arXiv:1409.2444 [math.GR], 2014.
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 [broken link]
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
Eric Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

Cf. A001423, A001426, A023813, A023815, A027851.

a(n) + A079172(n) = A002489(n).

a(8), a(9) from Distler and Kelsey (2013). - N. J. A. Sloane, Feb 19 2013

A023813 a(n) = n^(n*(n+1)/2).

Original entry on oeis.org

1, 1, 8, 729, 1048576, 30517578125, 21936950640377856, 459986536544739960976801, 324518553658426726783156020576256, 8727963568087712425891397479476727340041449, 10000000000000000000000000000000000000000000000000000000

Offset: 0

Lyle Ramshaw (ramshaw(AT)pa.dec.com)

Determinant of n X n matrix M_(i,j) = binomial(n*i,j). - Benoit Cloitre, Sep 13 2003

Number of commutative binary operations on an n-set. Labeled commutative groupoids.

Eric Postpischil, Posting to sci.math newsgroup, May 21 1990
Index entries for sequences related to groupoids

a(n) + A079182(n) = A002489(n).

Cf. A000217, A001425, A002489, A023814, A023815.

seq(mul(n^k, k=1..n), n=0..10); # Zerinvary Lajos, Jun 03 2007

Table[n^((n^2 + n)/2), {n, 1, 10}]  (* Geoffrey Critzer, Jan 27 2013 *)

a(n) = Product_{k=1..n} n^k. - José de Jesús Camacho Medina, Jul 12 2016

a(n) = n^A000217(n). - Alois P. Heinz, Aug 06 2018

Better description from Amarnath Murthy, Dec 29 2001

A079195 Number of non-associative commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 2, 666, 1047436, 30517547395, 21936950639192784, 459986536544739894613595, 324518553658426726783148869733112

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

Cf. A023813, A023815, A079192, A079196 (isomorphism classes), A079197, A079198.

A079192(n) + a(n) + A079198(n) + A023815(n) = A002489(n).
a(n) = Sum_{k>=1} A079197(n,k)*A079210(n,k).
a(n) = A023813(n) - A023815(n). - Andrew Howroyd, Jan 26 2022

a(0)=0 prepended and a(5)-a(8) added by Andrew Howroyd, Jan 26 2022

A079201 Number of isomorphism classes of associative commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

1, 1, 0, 3, 0, 0, 3, 9, 0, 0, 0, 3, 0, 0, 16, 39, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 15, 0, 4, 0, 103, 201, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 6, 0, 0, 4, 91, 0, 55, 0, 715, 1258, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12

Offset: 0

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

Number of elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!).

			Triangle T(n,k) begins:
  1;
  1;
  0, 3;
  0, 0, 3, 9;
  0, 0, 0, 3, 0, 0, 16, 39;
  0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 15, 0, 4, 0, 103, 201;

Andrew Howroyd, Table of n, a(n) for n = 0..217 (rows 0..8)
C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to semigroups

Row sums are A001426.

Cf. A023815, A027423 (row lengths), A079171, A079194, A079197, A079200, A058105.

A079194(n,k) + A079197(n,k) + A079200(n,k) + T(n,k) = A079171(n,k).
T(n, A027423(n)) = A058105(n).
A023815(n) = Sum_{k>=1} T(n,k)*A079210(n,k).

a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, Jan 26 2022

A079240 Number of associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 0, 48, 2344, 153000, 15875924, 7676692856, 148188196673360

Offset: 0

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

nonn

Cf. A079207, A079241 (isomorphism classes), A079230, A079232, A079234, A079236, A079195, A079242, A079244, A063524.

A079230(n) + A079232(n) + A079234(n) + A079236(n) + A079195(n) + a(n) + A079242(n) + A079244(n) + A063524(n) = A002489(n).
a(n) = Sum_{k>=1} A079207(n,k)*A079210(n,k).
a(n) = A023814(n) - A023815(n) - A079242(n). - Andrew Howroyd, Jan 27 2022

a(0)=0 prepended and a(5)-a(8) added by Andrew Howroyd, Jan 27 2022

A079198 Number of associative non-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 2, 50, 2352, 153002, 15876046, 7676692858

Offset: 1

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

a(n) + A079192(n) + A079195(n) + A023815(n) = A002489(n).
Each a(n) is equal to the sum of the products of each element in row n of A079200 and the corresponding element of A079210.
Since this is the number of labeled noncommutative semigroups on an n-set, a(n) = A023814(n)-A023815(n). - Stanislav Sykora, Apr 03 2016

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

Cf. A023814, A023815, A079192, A079195, A079199, A079200.

a(5)-a(7) added by Stanislav Sykora, Apr 03 2016

A079192 Number of non-associative non-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 6, 18904, 4293916368, 298023193359221998, 10314424798468598595515695154, 256923577521058877628624940679487983651948, 6277101735386680763835789098689112757675628513119817261598

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

Cf. A002489, A023813, A023814, A023815, A079193 (isomorphism classes), A079194, A079195, A079198.

a(n) + A079195(n) + A079198(n) + A023815(n) = A002489(n).
a(n) = Sum_{k>=1} A079194(n,k)*A079210(n,k).
a(n) = A002489(n) - A023813(n) - A023814(n) + A023815(n). - Andrew Howroyd, Jan 26 2022

a(0)=0 prepended and a(5)-a(8) added by Andrew Howroyd, Jan 26 2022

A058167 Triangle read by rows: T(n,k) is the number of labeled commutative semigroups of order n with k idempotents.

Original entry on oeis.org

1, 4, 2, 24, 30, 9, 260, 492, 312, 76, 4805, 10060, 9900, 4900, 1065, 157956, 284130, 348420, 259500, 112500, 22566, 12277440, 11892846, 14768775, 14093380, 9063600, 3592554, 674611, 3287166928, 896150920, 812261856, 854806120, 707722680, 413149464, 152565280, 27019896

Offset: 1

Christian G. Bower, Nov 15 2000

			Triangle begins:
     1;
     4,     2;
    24,    30,    9;
   260,   492,  312,   76;
  4805, 10060, 9900, 4900, 1065;
  ...

Index entries for sequences related to semigroups

Row sums give A023815.
Main diagonal is A058164(n+1).
Cf. A058116 (isomorphism classes).

a(29)-a(36) from Andrew Howroyd, Jan 27 2022

cp's OEIS Frontend

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

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A001426 Number of commutative semigroups of order n.

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A023814 Number of associative binary operations on an n-set; number of labeled semigroups.

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A023813 a(n) = n^(n*(n+1)/2).

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A079195 Number of non-associative commutative closed binary operations on a set of order n.

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A079201 Number of isomorphism classes of associative commutative closed binary operations on a set of order n, listed by class size.

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A079240 Number of associative non-commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.

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A079198 Number of associative non-commutative closed binary operations on a set of order n.

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A079192 Number of non-associative non-commutative closed binary operations on a set of order n.

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