I just saw a table which counts diffeomorphism classes of homotopy $n$-spheres (that is, spaces homotopy equivalent to $n$-dimensional spheres). Such a table can be seen on the first page of this paper. Most of these numbers are less than $10$, and all of them except $15$ is less than a thousand. But then $15$ has $16,256$ different classes.
I always thought of higher dimensional spheres as being fairly homogeneous as you went out, so this variance came as a surprise. My question is whether there is an easy way to explain such a large jump. Also, is there a more current table of values available?