  The polytope of integer partitions of $n$ is the convex hull of the
corresponding $n$-dimensional integer points. Its vertices are of importance
because every partition is their convex combination. Computation shows
intriguing features of $v(n),$ the number of the polytope vertices: its graph
has a saw-toothed shape with the highest peaks at prime $n$'s. We explain the
shape of $v(n)$ by the large number of partitions of even $n$'s that were
counted by N. Metropolis and P. R. Stein. These partitions are convex
combinations of two others. We reveal that divisibility of $n$ by 3 also
reduces the value of $v(n),$ which is caused by partitions that are convex
combinations of three but not two others, and characterize convex
representations of such integer points in arbitrary integral polytope. To
approach the prime $n$ phenomenon, we use a specific classification of integers
and demonstrate that the graph of $v(n)$ is stratified to layers corresponding
to resulting classes. Our main conjecture claims that $v(n)$ depends on
collections of divisors of $n.$ We also offer an initial argument for that the
number of vertices of the Gomory's master corner polyhedron on the cyclic group
has features similar to those of $v(n).$
