A079189 -id:A079189 - 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

A079190 Number of isomorphism classes of anti-commutative closed binary operations (groupoids) on a set of order n.

Original entry on oeis.org

1, 6, 996, 31857648, 266666713602640, 929809173755713574913480, 2002123402266181527640478418179038176, 3702236248557739850415303240942330019881771301360640, 7805296829528400289943264314587254996361382902046539931447903763389056

Offset: 1

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

nonn

A079187(n)+A079190(n)=A001329(n).

Each a(n) is equal to the sum of the elements in row n of A079191.

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

Cf. A079187, A079189, A079191.

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, ...] = Product_{i>=1, 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))^(s_i*(i*s_i+1)/2) * (-1 + Sum_{d|i} (d*s_d))^(s_i*(i*s_i-1)/2) or {i=j, even} (Sum_{d|i and i/d is odd} (d*s_d))^s_i * (Sum_{d|i} (d*s_d))^(i*s_i^2/2) * (-1 + Sum_{d|i} (d*s_d))^(s_i*(i*s_i-2)/2) or {i < j} (Sum_{d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j) or {i > j} (-1 + Sum_{d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j). [Corrected by Sean A. Irvine, Aug 03 2025]

a(n) is asymptotic to (n^binomial(n+1, 2) * (n-1)^binomial(n, 2))/n! = A079189(n)/A000142(n)

Edited, corrected and extended with formula by Christian G. Bower, Dec 12 2003

a(9) from Sean A. Irvine, Aug 03 2025

A079186 Number of non-anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 8, 13851, 3530555392, 266023223876953125, 9644962193498535546171949056, 246832875573638552740275218239438131202951, 6127827569844832702316847785612357470342156990019367075840, 193794664362053647720926884692597177807303542565053791345764052714030485961865

Offset: 0

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

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..20
C. van den Bosch, Closed binary operations on small sets
Index entries for sequences related to groupoids

Cf. A002489, A079187 (non-isomorphic), A079188, A079189, A079210.

a(n) = n^(n^2) - (n^n)*((n^2-n)^((n^2-n)/2)) \\ Andrew Howroyd, Jan 23 2022

a(n) = n^(n^2) - (n^n)*((n^2-n)^((n^2-n)/2)).
a(n) = A002489(n) - A079189(n).
a(n) = Sum_{k>=1} A079178(n,k)*A079210(n,k).

a(0)=0 prepended and terms a(5) and beyond from Andrew Howroyd, Jan 23 2022

A079191 Number of isomorphism classes of anti-commutative closed binary operations on a set of order n, listed by class size.

Original entry on oeis.org

1, 4, 2, 2, 8, 34, 952, 2, 1, 14, 6, 211, 283, 13570, 31843561

Offset: 1

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

A079188(n)+A079191(n)=A079171(n).
Elements per row: 1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)
First four rows: 1; 4,2; 2,8,34,952; 2,1,14,6,211,283,13570,31843561
A079189(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 x of this sequence is given by A079190(x).

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

Cf. A079188, A079189, A079190.

cp's OEIS Frontend

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

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A079190 Number of isomorphism classes of anti-commutative closed binary operations (groupoids) on a set of order n.

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A079186 Number of non-anti-commutative closed binary operations on a set of order n.

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A079191 Number of isomorphism classes of anti-commutative closed binary operations on a set of order n, listed by class size.

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