How many non-isomorphic semigroups are there of orders $2$ and $3$?

Question

For order $2$, I have found 5. There are 16 maps from $\{a, b\} \times \{a, b\} \to\{a, b\}$. They form $10$ equivalence classes of non-isomorphic binary operations, $5$ of which are associative. Of these $2$ are monoids and $1$ is a group.They are

1. The group $ \mathrm Z/2 $
2. $\{ 0, 1\}$ with standard multiplication.
3. $\{a, b\}$ with $x \circ y = a$ for all $x$ and $y$ $=a$ or $b$
4. $\{a, b\}$ with $x \circ y = x$ for all $x$ and $y$ $=a$ or $b$
5. $\{a, b\}$ with $x \circ y = y$ for all $x$ and $y$ $=a$ or $b$

Is $5$ the correct number of semigroups for $n=2$ ?

How many semigroups are there for $n=3$ ? No proof is needed, but a reference would be appreciated.

Answer 1

The reference is http://oeis.org/A027851

There are $5$ nonisomorphic semigroups of order $2,$ and $24$ of order $3.$

Answer 2

Up to isomorphism and anti-isomorphism, there are 4 semigroups of order 2, and 18 of order 3. (An anti-isomorphism is essentially transposing the multiplication table, up to re-labeling; formally, an anti-isomorphism of semigroups is a bijection $\phi: S\rightarrow T$ where $\phi(s_1s_2)=\phi(s_2)\phi(s_1)$.)

Indeed, up to isomorphism and anti-isomorphism: $$ \begin{array}{c|r} \text{Size}&\text{Number of semigroups}\\ \hline 1& 1\\ 2& 4\\ 3& 18\\ 4& 126\\ 5& 1,160\\ 6& 15,973\\ 7& 836,021\\ 8& 1,843,120,128\\ 9& 52,989,400,714,478 \end{array} $$ For a reference, you can find this table in the documentation for the SmallSemi GAP package. (Like the Small Groups package, but for semigroups.)