Suppose $A$ is a set, $f:A^2 \mapsto A$. We call $f$ associative iff $\forall x, y, z \in A$ $f(x, f(y, z)) = f(f(x, y), z)$.
Now, suppose $|A| = n$. How many associative functions are there on $A$? I would like to know the asymptotics of their number as $n \to \infty$.
Despite the question seems quite natural, the most relevant thing I was able to find was OEIS A027851 - the number of $n$-element semigroups up to isomorphism. However, it is not what I need, as two different associative functions may nevertheless yield isomorphic semigroups.
