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

0, 4, 23, 186, 1914, 28632, 1627671, 3684030412, 105978177936290

Offset

1,2

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.

Links

Table of n, a(n) for n=1..9.

Eric W. Weisstein, Finite Group

Eric W. Weisstein, Semigroup

Formula

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

Example

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.

Crossrefs

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

Sequence in context: A302189 A292971 A317967 * A108953 A369140 A222454

Adjacent sequences: A186114 A186115 A186116 * A186118 A186119 A186120

Keyword

nonn,hard,less

Author

Jonathan Vos Post, Feb 13 2011

Status

approved