TITLE: How many finite loops?
QUESTION [7 upvotes]: How many finite loops  of order $n$ are there? 
I am interested in the exact values ​​of $n$ if $n <40$ or even reasonable estimates. I am also interested in formulae or bounds for all $n$.
Note that similar questions have been studied for groups and semigroups.  Much information is available in the group theory case, but even for semigroups not much is known beyond $n=11$.

REPLY [2 votes]: Here you can find the number of loops of order n up to isotopy and isomorphism for n at most 10, and the number of quasigroups up to isomorphism.
http://cs.anu.edu.au/~bdm/papers/ls_final.pdf