A352622 Number of regular convex polytopes that can be formed with n indistinguishable points located at the vertices, coinciding in equal frequency at each vertex, if coinciding at all.

1, 2, 2, 4, 3, 6, 3, 8, 4, 7, 3, 12, 3, 7, 6, 12, 3, 11, 3, 13, 6, 7, 3, 20, 5, 7, 6, 12, 3, 16, 3, 16, 6, 7, 7, 20, 3, 7, 6, 20, 3, 16, 3, 12, 10, 7, 3, 27, 5, 12, 6, 12, 3, 16, 7, 19, 6, 7, 3, 29, 3, 7, 10, 20, 7, 16, 3, 12, 6, 17, 3, 31, 3, 7, 10, 12, 7, 16

Offset: 1

Author

Rajan Murthy, Mar 24 2022

Keywords

• nonn

Comments

For n = 1: there is only a 0-dimensional simplex.

For n = 2: the two points may coincide or may form a 1-dimensional simplex.

For n = 3: the three points may coincide or may form a 2-dimensional simplex.

For n = 2^(k+1), where k is a positive integer: a(n) = k + (k+2) + (k-1) + (k-1) = 4*k: k polygons (one for each factor > 2), k+2 simplexes (one for each factor), k-1 cubes (one for each even factor > 4, the cubes for 2 and 4 are a simplex and polygon, respectively), and k-1 orthoplexes (one for each even factor > 4, orthoplexes with 1, 2, and 4 vertices are already counted).

For prime numbers greater than 3 (n = p > 3, where p is prime): a(n) is always 3:

(1) the 0-dimensional polytope (all points coinciding), (2) a 2-dimensional p-gon, where p is a prime n, and (3) a (p-1)-dimensional simplex.

For even numbers which are not powers of 2: a(n) = 2*(number of factors) + (number of even factors) - 3 + adjustments. The adjustments are as follows: -1 if n is a multiple of 3; -1 if n is a multiple of 4; +1 for each positive integer k such that 2^(k+2) is a factor of n; +1 for each factor of n which is in the set (12,20,24,120,600). With the exception of factors 1 and 2, every factor contributes a simplex and a polygon. Even factors add a third polytope which is an orthoplex. Factors 1 and 2 only add a zero-dimensional and one-dimensional simplex respectively and so a total of three is subtracted (-1 for each of factors 1 + 2 and -1 for the even factor 2). The polygon and the simplex to which the factor of 3 maps are identical leading to an adjustment of -1. The polygon and the 2-dimensional "cube" that a factor of 4 maps to are identical also leading to a -1 adjustment. Factors which are powers of 2 greater than 4 and factors which correspond to a polytope peculiar to 3 or 4 dimensions each add one more possible polytope.

For nonprime odd numbers which are multiples of 3: a(n) = 2*(the number of factors) - 2. Each factor maps to a polygon and a simplex, but for the factor 3 the polygon is the simplex, and the factor 1 maps to a single coincident point.

For nonprime odd numbers which are not multiples of 3: a(n) = 2*(the number of factors) - 1. Each factor > 1 maps to a polygon and a simplex and the factor 1 maps to a single coincident point.

Examples

			For n = 12, the set of factors of 12 is (1, 2, 3, 4, 6, 12): 2 odd and 4 even including adjusting factors (3, 4, and 12). a(n) = 2*2 + 3*4 - 3 - 1 - 1 + 1 = 12: (1) a 0-dimensional simplex with 12 coincident points; (2) a 1-dimensional simplex with 2 groups of 6 coincident points; (3) a 2-dimensional simplex with 3 groups of 4 coincident points; (4,5) a square and a 3-dimensional simplex each with 4 groups of 3 coincident points; (6,7,8) a hexagon, an octahedron, and a 5-dimensional simplex each with 2 coincident points at the vertices; (9, 10, 11, 12) a dodecagon, a 6-dimensional orthoplex, an 11-dimensional simplex, and an icosahedron each with no coincident points.
For n = 20, the set of factors of 20 is (1, 2, 4, 5, 10, 20): 2 odd and 4 even including adjusting factors (4 and 20). a(n) = 2*2 + 3*4 - 3 - 1 + 1 = 13: (1) a 0-dimensional simplex with 20 coincident points; (2) a 1-dimensional simplex with 2 groups of 10 coincident points; (3, 4) a square and a 3-dimensional simplex each with 4 groups of 5 coincident points; (5, 6) a pentagon, and a 4-dimensional simplex each with groups of 4 coincident points; (7, 8, 9) a decagon, a 5-dimensional orthoplex, and a 9-dimensional simplex each with 2 coincident points at the vertices; (10, 11, 12, 13) a 20-sided polygon, a 10-dimensional orthoplex, a 19-dimensional simplex, and a dodecahedron.
For n = 24, the set of factors of 24 is (1, 2, 3, 4, 6, 8, 12, 24): 2 odd and 6 even including adjusting factors (3, 4, 8, 12, and 24). a(n) = 2*2 + 3*6 - 3 - 1 - 1 + 1 + 1 + 1 = 20: (1) a 0-dimensional simplex with 24 coincident points; (2) a 1-dimensional simplex with 2 groups of 12 coincident points; (3) a 2-dimensional simplex with 3 groups of 8 coincident points; (4, 5) a square and a 3-dimensional simplex each with 4 groups of 6 coincident points; (6, 7, 8) a hexagon, an octahedron, and a 5-dimensional simplex each with 4 coincident points; (9, 10, 11, 12) an octagon, a cube, a 4-dimensional orthoplex, a 7-dimensional simplex each with 3 coincident points; (13, 14, 15, 16) a dodecagon, a 6-dimensional orthoplex, an 11-dimensional simplex, and an icosahedron each with 2 coincident points; (17, 18, 19, 20) a 24-sided polygon, a 4-dimensional 24-cell, a 12-dimensional orthoplex, and a 23-dimensional simplex.
		

References

• E. W. Weisstein, CRC Encyclopedia of Mathematics, 3rd Ed., CRC Press, 2009, 3037-3038.

Crossrefs

Cf. A053016, A063723, A063927, A111336.

Formula

a(n) = Sum_{i|n} A111336(i).