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.
%I A186117 #9 Feb 16 2025 08:33:14 %S A186117 0,4,23,186,1914,28632,1627671,3684030412,105978177936290 %N A186117 Number of nonisomorphic semigroups of order n minus number of groups of order n. %C A186117 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. %H A186117 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/FiniteGroup.html">Finite Group</a> %H A186117 Eric W. Weisstein, <a href="https://mathworld.wolfram.com/Semigroup.html">Semigroup</a> %F A186117 a(n) = A027851(n) - A000001(n). %e A186117 a(1) = 0 because there are unique groups and semigroups of order 1, so 1 - 1 = 0. %e A186117 a(2) = 4 because there are 5 semigroups of order 2 groups and a unique group of order 2, so 5 - 1 = 4. %Y A186117 Cf. A000001, A027851, A001423, A029851, A001426, A023814, A058108, A079173, A001329, A186116. %K A186117 nonn,hard,less %O A186117 1,2 %A A186117 _Jonathan Vos Post_, Feb 13 2011