Matrygons timer 20s memory 1024MB

A matryoshka is a type of doll that originated in Russia over a century ago. Their defining characteristic is that they consist of a set of dolls, all of a different size, with smaller dolls fitting nicely inside larger dolls.

In this problem, we work with matrygons, which are sets of regular convex polygons that follow a similar nesting pattern. A matrygon consists of a set of regular convex polygons with positive area p_1, p_2, \dots, p_k such that, for all i, the vertices of p_{i+1} overlap with a proper subset of the vertices of p_i (p_{i+1} has strictly less vertices than p_i).

For example, the following pictures illustrates two matrygons. The first one contains 3 regular convex polygons: a regular icositetragon (24 sides), a regular hexagon (6 sides), and an equilateral triangle (3 sides). The second one contains 2 regular convex polygons: a regular icosidigon (22 sides) and a regular hendecagon (11 sides). Each of these matrygons has 33 total sides among all polygons in it.

Given a fixed total number of sides \mathbf{N}, calculate the largest number of polygons that can be part of a matrygon such that the total number of sides among all polygons in it is exactly \mathbf{N}.