cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A079245 Number of isomorphism classes of associative commutative non-anti-associative non-anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 3, 12, 58
Offset: 1

Views

Author

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

Keywords

Comments

A079231(n)+A079233(n)+A079235(n)+A079237(n)+A079196(n)+A079241(n)+A079243(n)+A079245(n)+A063524(n)=A002489(n)
Each a(n) is equal to the sum of the elements in row n of A079209.

Links

Crossrefs

Cf. A079209, A079244, A079231, A079233, A079235, A079237, A079196, A079241, A079243, A063524.

A079241 Number of isomorphism classes 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, 10, 127, 1588, 26487, 1610379, 3683808608
Offset: 0

Views

Author

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

Keywords

Links

Crossrefs

Row sums of A079207.
Cf. A079240 (labeled case), A079231, A079233, A079235, A079237, A079196, A079243, A079245, A063524.

Formula

A079231(n) + A079233(n) + A079235(n) + A079237(n) + A079196(n) + a(n) + A079243(n) + A079245(n) + A063524(n) = A002489(n).
a(n) = A027851(n) - A001426(n) - A079243(n). - Andrew Howroyd, Jan 27 2022

Extensions

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

A079243 Number of isomorphism classes of associative non-commutative non-anti-associative anti-commutative closed binary operations on a set of order n.

Original entry on oeis.org

0, 0, 2, 2, 3, 2, 4, 2, 4
Offset: 0

Views

Author

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

Keywords

Comments

The only closed binary operations that are both commutative and anti-commutative are those on sets of size <= 1. The significance of non-commutative (and non-anti-associative) in the name is that it excludes this possibility. Otherwise, the first two terms would be 1. - Andrew Howroyd, Jan 26 2022

Examples

			From _Andrew Howroyd_, Jan 26 2022: (Start)
The a(6) = 4 operations are the two shown below and their converses.
    | 1 2 3 4 5 6         | 1 2 3 4 5 6
  --+------------       --+------------
  1 | 1 2 3 4 5 6       1 | 1 2 3 1 2 3
  2 | 1 2 3 4 5 6       2 | 1 2 3 1 2 3
  3 | 1 2 3 4 5 6       3 | 1 2 3 1 2 3
  4 | 1 2 3 4 5 6       4 | 4 5 6 4 5 6
  5 | 1 2 3 4 5 6       5 | 4 5 6 4 5 6
  6 | 1 2 3 4 5 6       6 | 4 5 6 4 5 6
(End)

Links

Crossrefs

Row sums of A079208.
Cf. A079242 (labeled case), A079231, A079233, A079235, A079237, A079196, A079241, A079245, A063524.

Formula

A079231(n) + A079233(n) + A079235(n) + A079237(n) + A079196(n) + A079241(n) + a(n) + A079245(n) + A063524(n) = A002489(n).
Conjecture: a(n) = A000005(n) for n > 1. - Andrew Howroyd, Jan 26 2022

Extensions

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

A118100 Number of commutative semigroups of order <= n.

Original entry on oeis.org

1, 2, 5, 17, 75, 400, 2543, 19834, 241639, 11787482, 3530717819
Offset: 0

Views

Author

Jonathan Vos Post, May 11 2006

Keywords

Comments

A001426(n) is the number of commutative semigroups of order n. A001426(n) + A079193(n) + A079196(n) + A079199(n) = A001329(n). 2, 5, 17, 2543 and 241639 are primes.

Examples

			a(8) = 1 + 1 + 3 + 12 + 58 + 325 + 2143 + 17291 + 221805 = 241639.

Links

Crossrefs

Cf. A001423, A001426, A023815, A027851, A058105, A058116.

Formula

a(n) = Sum_{i=1..n} A001426(i).

Extensions

a(9)-a(10) added using the terms in A001426 by Miles Englezou, May 27 2025

A118542 Number of nonisomorphic groupoids with <= n elements.

Original entry on oeis.org

1, 2, 12, 3342, 178985294, 2483527716080119, 14325590005802419238355799, 50976900301828909677297289506452525838, 155682086691137998248942804080553139214788341933547854
Offset: 0

Views

Author

Jonathan Vos Post, May 06 2006

Keywords

Comments

The number of isomorphism classes of closed binary operations on sets of order <= n. See formulas by Christian G. Bower in A001329 Number of nonisomorphic groupoids with n elements.

Examples

			a(5) = 1 + 1 + 10 + 3330 + 178981952 + 2483527537094825 = 2483527716080119 is prime.

Crossrefs

Cf. A001424, A001425, A002489, A006448, A029850, A030245-A030265, A030271, A038015-A038023.

Formula

a(n) = SUM[i=0..n] A001329(i). a(n) = SUM[i=0..n] (A079173(i)+A027851(i)). a(n) = SUM[i=0..n] (A079177(i)+A079180(i)). a(n) = SUM[i=0..n] (A079183(i)+A001425(i)). a(n) = SUM[i=0..n] (A079187(i)+A079190(i)). a(n) = SUM[i=0..n] (A079193(i)+A079196(i)+A079199(i)+A001426(i)).
Previous Showing 11-15 of 15 results.

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