Edited by _N. J. A. Sloane, _, Jul 25 2009

_Jonathan Vos Post (jvospost3(AT)gmail.com), _, May 08 2006

One more term from _Jonathan Vos Post (jvospost3(AT)gmail.com), _, Jul 20 2009

Number of monoids (semigroups with identity) of order <= n.

Partial sums of A058129.

1, 3, 10, 45, 273, 2510, 34069, 1703066

Monoid analogue of A063756 Number of groups of order <= n. See also A118581 Number of nonisomorphic semigroups of order <= n. A semigroup is an algebraic structure consisting of a set S closed under an associative binary operation (and thus is an associative groupoid). Some sources require that a semigroup have an identity element (in which case semigroups are identical to monoids). Not all sources agree that S should be nonempty. A118581 assumes that a semigroup may be empty and need not have an identity. This sequence, however, requires an identity and thus cannot be empty.

hard,nonn,new

One more term from Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 20 2009

Edited by N. J. A. Sloane, Jul 25 2009

Monoid analogue of A063756 Number of groups of order <= n. See also A118581 Number of nonisomorphic semigroups of order <= n. A semigroup is an algebraic structure consisting of a set S closed under an associative binary operation (and thus is an associative groupoid). Some sources require that a semigroup have an identity element (in which case semigroups are identical to monoids). Not all sources agree that S should be nonempty. A118581 assumes that a semigroup may be empty, and need not have an identity. This sequence, however, requires an identity, and thus cannot be empty.

hard,nonn,new

hard,nonn,new

Jonathan Vos Post (jvospost2jvospost3(AT)yahoogmail.com), May 08 2006

Number of monoids (semigroups with identity) of order <= n.

1, 3, 10, 45, 273, 2510, 34069

1,2

Monoid analogue of A063756 Number of groups of order <= n. See also A118581 Number of nonisomorphic semigroups of order <= n. A semigroup is an algebraic structure consisting of a set S closed under an associative binary operation (and thus is an associative groupoid). Some sources require that a semigroup have an identity element (in which case semigroups are identical to monoids). Not all sources agree that S should be nonempty. A118581 assumes that a semigroup may be empty, and need not have an identity. This sequence, however, requires an identity, and thus cannot be empty.

a(n) = SUM[i=1..n] A058129(i). a(n) = SUM[i=1..n] (2*A058133(i) - A058132(i)).

Cf. A001329, A001423, A001426, A023814, A027851, A029851, A058108, A058132, A058133, A063756, A079173, A118581.

hard,nonn,new

Jonathan Vos Post (jvospost2(AT)yahoo.com), May 08 2006

approved