A079182 -id:A079182 - 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

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

A079184 Number of isomorphism classes of non-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

0, 4, 2, 2, 8, 70, 3121, 2, 1, 14, 22, 275, 467, 48306, 178888897

Offset: 1

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

A079184(n)+A079185(n)=A079171(n).
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 0; 4,2; 2,8,70,3121; 2,1,14,22,275,467,48306,178888897
A079176(x) is equal to the sum of the products of each element in row x of this sequence and the corresponding element of A079210.
The sum of each row n is given by A079177(n).

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

Cf. A079182, A079183, A079185.

cp's OEIS Frontend

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

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

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A079184 Number of isomorphism classes of non-commutative closed binary operations on a set of order n, listed by class size.

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