The parity of the number of quasigroups

Let I ( n ) be the number of isomorphism classes of quasigroups of order n . Despite prior enumerations showing that I ( n ) is odd for 1 ≤ n ≤ 11 , we find that I ( 12 ) is even. We also give a method for finding the parity of I ( n ) , which we use to show that I ( n ) is odd for n ∈ { 13 , 14 , 15 , 16 , 17 , 19 , 21 } .