I am considering the collection of all magmas (sets with binary operations) of order 3. Since we just need a binary operation and no other properties, it makes sense to define a magma in terms of all of its binary products:
$a*a, a*b, a*c, b*a, b*b, b*c, c*a, c*b, c*c$.
So we have 9 products. Since we require no other properties, any of $a, b, c$ is a valid value for any product. So to determine our magma, we make 9 choices from a set of 3. So it seems like we have $3^9$ possible magmas of order 3.
However, some of these magmas are identical. For example, if we have a magma defined by our 9 equations, and we just switch the roles of $a$ and $b$ in every equation, we haven't changed the magma at all. We've just reordered our equations and relabeled our elements, but the algebraic structure is the same. So by swapping $a$, $b$, and $c$, we keep our magma the same.
The amount of permutations of a list of length 3 is $3!$. So it seems like a given magma should have $3!$ ways of being defined by our 9 equation method.
So this leads us to the conclusion that we have $\frac {3^9} {3!}$ possible magmas of order 3. The only issue is that this is not an integer. Obviously, a non-integer value for the number of possible magmas does not really make sense, so what have I done wrong here?
This idea can also be extended to the number of magmas of order n. Using similar logic would lead us to believe that it should be $\frac {n^{n^2}} {n!}$, but in general, this is not an integer.
So what is the correct way to find the number of magmas of order n? If a general method doesn't exist or is overly complicated, then I am fine with focusing on the $n = 3$ case.
I wrote a program to apply Burnside's lemma to determine the number of nonisomorphic magmas of order $n$. Turn's out its in OEIS. Linked there is this reference specifically on the $n=3$ case.