A023813 -id:A023813 - 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

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.

A023815 Number of binary operations on an n-set that are commutative and associative; labeled commutative semigroups.

Original entry on oeis.org

1, 1, 6, 63, 1140, 30730, 1185072, 66363206, 7150843144, 3829117403448

Offset: 0

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

Eric Postpischil Posting to sci.math newsgroup, May 21 1990
Amit Sehgal, Sunil Kumar, Sarita, Yashpal, Commutative Associative Binary Operations on a Set with Five Elements, J. Phys.: Conf. Ser. (2018) 1000 012063
Eric Weisstein's World of Mathematics, Semigroup.
Index entries for sequences related to semigroups

Row sums of A058167.
Cf. A001423, A001426 (isomorphism classes), A023813 (commutative only), A023814 (associative only), A027851.
Cf. A002489, A079192, A079195, A079198, A079201, A079210.

a(n) + A079192(n) + A079195(n) + A079198(n) = A002489(n).

a(n) = Sum_{k>=1} A079201(n,k)*A079210(n,k). - Andrew Howroyd, Jan 26 2022

a(8) from Andrew Howroyd, Jan 26 2022

a(9) from Andrew Howroyd, Feb 14 2022

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

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

A079189 Number of anti-commutative closed binary operations (groupoids) on a set of order n.

Original entry on oeis.org

1, 1, 8, 5832, 764411904, 32000000000000000, 669462604992000000000000000, 10090701947420325348336258984797490118656, 149274165541848061518941637595308945760198454444667437056, 2832386113499265897149023834314938475799908379160975581551362823935905234944

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. A023813, A079186, A079190 (isomorphism classes), A079191, A079210.

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

a(n) = (n^n)*((n^2-n)^((n^2-n)/2)).
a(n) = A002489(n) - A079186(n).
a(n) = Sum_{k>=1} A079191(n,k)*A079210(n,k).
a(n) = A023813(n)*A023813(n-1).

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

a(0)=1 prepended, a(8) corrected and a(9) added by Andrew Howroyd, Jan 23 2022

A079182 Number of non-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 8, 18954, 4293918720, 298023193359375000, 10314424798468598595531571200, 256923577521058877628624940679495660344806, 6277101735386680763835789098689112757675628661308013936640

Offset: 1

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

nonn

A023813(n) + A079182(n) = A002489(n).

Each a(n) is equal to the sum of the products of each element in row n of A079184 and the corresponding element of A079210.

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

Cf. A023813, A079178, A079180.

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

a(n) = n^(n^2)-n^((n^2-n)/2).

More terms from Geoffrey Critzer, Jan 27 2013

A346061 A(n,k) = n! * [x^n] (Sum_{j=0..n} k^(j(j+1)/2) x^j/j!)^(1/k) if k>0, A(n,0) = 0^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 1, 0, 1, 1, 7, 23, 1, 0, 1, 1, 13, 199, 393, 1, 0, 1, 1, 21, 901, 17713, 13729, 1, 0, 1, 1, 31, 2861, 249337, 4572529, 943227, 1, 0, 1, 1, 43, 7291, 1900521, 264273961, 3426693463, 126433847, 1, 0

Offset: 0

Alois P. Heinz, Jul 03 2021

A(n,k) is odd if k >= 1 or n = 0.

			Square array A(n,k) begins:
  1, 1,     1,       1,         1,          1, ...
  0, 1,     1,       1,         1,          1, ...
  0, 1,     3,       7,        13,         21, ...
  0, 1,    23,     199,       901,       2861, ...
  0, 1,   393,   17713,    249337,    1900521, ...
  0, 1, 13729, 4572529, 264273961, 6062674201, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..43, flattened
Richard Stanley, Proof of the general conjecture, MathOverflow, March 2021.

Columns k=0-3 give: A000007, A000012, A178315, A178319.
Rows n=0-2 give: A000012, A057427, A002061 (for k>0).
Main diagonal gives A342578.
Cf. A000217, A023813.

A:= (n, k)-> `if`(k>0, n!*coeff(series(add(k^(j*(j+1)/2)*
             x^j/j!, j=0..n)^(1/k), x, n+1), x, n), k^n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

E.g.f. of column k>0: (Sum_{j>=0} k^(j*(j+1)/2) * x^j/j!)^(1/k).
E.g.f. of column k=0: 1.
A(n,k) == 1 (mod k*(k-1)) for k >= 2 (see "general conjecture" in A178319 and link to proof by Richard Stanley above).

A342578 a(n) = n! * [x^n] (Sum_{j>=0} n^(j(j+1)/2) x^j/j!)^(1/n) for n > 0, a(0) = 1.

Original entry on oeis.org

1, 1, 3, 199, 249337, 6062674201, 3653786369479951, 65709007885111803731947, 40564683796482484146182142025377, 969773549559254966290998252899999751714721, 999999990999996719397362087568018696141879478712251051, 49037072510879011742983689973641327840345400616866967292640434759551

Offset: 0

Alois P. Heinz, Mar 15 2021

nonn

All terms are odd.

Alois P. Heinz, Table of n, a(n) for n = 0..35
Richard Stanley, Proof of the general conjecture, MathOverflow, March 2021.

Cf. A000217, A023813, A178315, A178319.

Main diagonal of A346061.

a:= n-> `if`(n>0, coeff(series(add(n^binomial(j+1, 2)*
         x^j/j!, j=0..n)^(1/n), x, n+1), x, n)*n!, 1):
seq(a(n), n=0..12);

a(n) == 1 (mod n*(n-1)) for n >= 2 (see "general conjecture" in A178319 and link to proof by Richard Stanley above).

a(n) ~ n^((n^2 + n - 2)/2). - Vaclav Kotesovec, Jul 15 2021

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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A001425 Number of commutative groupoids with n elements.

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A023815 Number of binary operations on an n-set that are commutative and associative; labeled commutative semigroups.

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A079195 Number of non-associative 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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A079189 Number of anti-commutative closed binary operations (groupoids) on a set of order n.

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A079182 Number of non-commutative closed binary operations on a set of order n.

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A346061 A(n,k) = n! * [x^n] (Sum_{j=0..n} k^(j*(j+1)/2) * x^j/j!)^(1/k) if k>0, A(n,0) = 0^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A346061 A(n,k) = n! * [x^n] (Sum_{j=0..n} k^(j(j+1)/2) x^j/j!)^(1/k) if k>0, A(n,0) = 0^n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

A342578 a(n) = n! * [x^n] (Sum_{j>=0} n^(j(j+1)/2) x^j/j!)^(1/n) for n > 0, a(0) = 1.