A079172 -id:A079172 - 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

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

A079173 Number of isomorphism classes of non-associative closed binary operations on a set of order n.

Original entry on oeis.org

0, 5, 3306, 178981764, 2483527537092910, 14325590003318891522247046, 50976900301814584087291487087212542367, 155682086691137947272042502251643461917498831796991599, 541851439802559836957713164869818405872834954135521300809796661279574643

Offset: 1

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

nonn

Each a(n) is equal to the sum of the elements in row n of A079174.

Christian van den Bosch, Closed binary operations on small sets. [Wayback Machine link]
Index entries for sequences related to groupoids.

Cf. A001329, A027851, A079172, A079174.

a(n) = A001329(n) - A027851(n).

More terms from Christian G. Bower, Nov 26 2003

a(8)-a(9) added using the data at A001329 and A027851 by Amiram Eldar, Jul 19 2025

A079174 Number of isomorphism classes of non-associative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 2, 3, 1, 12, 71, 3222, 0, 1, 14, 23, 270, 495, 48748, 178932213

Offset: 1

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

A079174(n)+A079175(n)=A079171(n)
Elements per row: 1,2,4,8,16,30,60,96 (A027423, number of positive divisors of n!)
First four rows: 0; 2,3; 1,12,71,3222; 0,1,14,23,270,495,48748,178932213
The sum of each row n is given by A079173(n).

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

Cf. A079172, A079173, A079175.

cp's OEIS Frontend

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

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

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A079173 Number of isomorphism classes of non-associative closed binary operations on a set of order n.

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A079174 Number of isomorphism classes of non-associative closed binary operations on a set of order n, listed by class size.

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