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 $v(n).$