OFFSET
1,4
COMMENTS
The minimum number of hyperfaces required for a convex polytope is 3, and the only convex polytope with 3 hyperfaces is a triangle. - Jianing Song, Sep 17 2018
From Rajan Murthy, Apr 08 2022: (Start)
For n = 1,2,3 point(s) can only be arranged in (n-1)-dimensional simplices. a(1)=a(2)=a(3) = 1. For n = 1,2,3 the point(s) can only be arranged in (n-1)-dimensional simplices. a(1)=a(2)=a(3) = 1.
For higher n, the figures based on n vertices are duals of the figures based on n hyperfaces. (End)
LINKS
FORMULA
a(3) = 1; a(n) = 2 if n = 4 or n is odd and >= 5; a(n) = 4 if n = 12, 20, 24, 120, 600 or a power of 2 >= 8; a(n) = 3 otherwise. - Jianing Song, Sep 17 2018
EXAMPLE
a(8) = 4 because the regular polytopes with 8 faces are the octagon, the octahedron, the four-dimensional cube and the 7-dimensional simplex.
From Rajan Murthy, Apr 08 2022: (Start)
For n = 8, points may be arranged in an octagon, a cube, a 4-dimensional orthoplex, or a 7-dimensional simplex, so a(8) = 4.
For n = 12, there are a(12) = 4 regular polytopes with 12 hyperfaces. They, and their duals with 12 points, are:
12 hyperfaces 12 points
dodecagon dodecagon
dodecahedron icosahedron
6-cube 6-D orthoplex
11-D simplex 11-D simplex
(End)
PROG
(PARI)
a(n)={
if(n<=3, return(1));
if(n==4||(n>=5&&n%2==1), return(2));
if(n>=6&&n%2==0, return(3+(n==12||n==20||n==24||n==120||n==600||(n>=8&&omega(2*n)==1))));
else(return(0));
} \\ Jianing Song, Sep 17 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paulo de A. Sachs (sachs6(AT)yahoo.de), Nov 09 2005
EXTENSIONS
Terms beyond a(38) from Jianing Song, Sep 17 2018
a(1) and a(2) prepended and definition extended by Rajan Murthy, Apr 08 2022
STATUS
approved