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.

Previous Showing 21-26 of 26 results.

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

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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)).

A186117 Number of nonisomorphic semigroups of order n minus number of groups of order n.

Original entry on oeis.org

0, 4, 23, 186, 1914, 28632, 1627671, 3684030412, 105978177936290
Offset: 1

Views

Author

Jonathan Vos Post, Feb 13 2011

Keywords

Comments

In a sense, this measures the increase in combinatorial structures available by dropping the requirement of inverses, and an identity element, in moving from the group axioms to the semigroup axioms. A semigroup is mathematical object defined for a set and a binary operator in which the multiplication operation is associative. No other restrictions are placed on a semigroup; thus a semigroup need not have an identity element and its elements need not have inverses within the semigroup. Other sequences may be derived by considering commutative semigroups and commutative groups, self-converse semigroup, counting idempotents, and the like.

Examples

			a(1) = 0 because there are unique groups and semigroups of order 1, so 1 - 1  = 0.
a(2) = 4 because there are 5 semigroups of order 2 groups and a unique group of order 2, so 5 - 1  = 4.

Links

Crossrefs

Cf. A000001, A027851, A001423, A029851, A001426, A023814, A058108, A079173, A001329, A186116.

Formula

a(n) = A027851(n) - A000001(n).

A209412 Number of nonisomorphic semigroups of order 2^n.

Original entry on oeis.org

1, 5, 188, 3684030417
Offset: 0

Views

Author

Jonathan Vos Post, Mar 08 2012

Keywords

Comments

This is to A000679 as semigroups are to groups.

Examples

			a(3) = 3684030417 because there are 3684030417 nonisomorphic semigroups of order 2^3 = 8.

Crossrefs

Cf. A000079, A000679 Number of groups of order 2^n, A027851 Number of nonisomorphic semigroups of order n.

Formula

a(n) = A027851(A000079(n)).

A318987 Number of semigroups of order n without identity.

Original entry on oeis.org

1, 0, 3, 17, 153, 1687, 26397, 1596113, 3682361420
Offset: 0

Views

Author

Steve Szabo, Sep 06 2018

Keywords

Links

Crossrefs

Cf. A027851, A058129.

Formula

a(n) = A027851(n) - A058129(n).

Extensions

More terms from Joerg Arndt, Dec 09 2018
a(0) prepended by Jianing Song, Oct 26 2019

A328746 Number of loops of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator).

Original entry on oeis.org

0, 1, 1, 1, 2, 5, 72, 12151, 53146457
Offset: 0

Views

Author

Jianing Song, Oct 26 2019

Keywords

Crossrefs

For the number of group-like algebraic structures of order n, see:
Semigroups: A027851 or A001423 (commutative: A001426);
Monoids: A058129 or A058133 (commutative: A058131);
Quasigroups: A057991 or A058171 (commutative: A057992);
Loops: A057771 or this sequence (commutative: A089925);
Groups: A000001 (commutative: A000688);
Rings: A027623 or A038036 (commutative: A037289);
Rings with unity: A037291;
Fields: A069513.

Formula

a(n) = (A057771(n)+A057996(n))/2.
Previous Showing 21-26 of 26 results.

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