The number of edges on the face of each Platonic solid is a divisor of the total number of edges (A063722) of its corresponding solid. The ratios of the total number of edges to face edges are: 6:3, 12:4, 12:3, 30:5, 30:3 --> giving the integer sequence 2, 3, 4, 6, 10.
Although a(n) is also the number of vertices on each face of the Platonic solids, they are not altogether divisors of the total number of vertices (A063723) with the tetrahedron as the only exception. The ratios are: 4:3, 8:4, 6:3, 20:5, 12:3.
Please see A053016 for an extensive list of web-resources about the Platonic Solids.