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%I #56 Jul 16 2022 01:37:02
%S 1,1,1,2,2,3,2,4,2,3,2,4,2,3,2,4,2,3,2,4,2,3,2,4,2,3,2,3,2,3,2,4,2,3,
%T 2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,4,2,3,2,3,
%U 2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2,3,2
%N Number of convex regular polytopes with n hyperfaces (n>2) or n vertices.
%C 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
%C From _Rajan Murthy_, Apr 08 2022: (Start)
%C 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.
%C For higher n, the figures based on n vertices are duals of the figures based on n hyperfaces. (End)
%H Wikipedia, <a href="https://it.wikipedia.org/wiki/Lista_dei_politopi_regolari">Lista dei politopi regolari</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/List_of_regular_polytopes">Regular Polytopes</a>.
%F 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
%e a(8) = 4 because the regular polytopes with 8 faces are the octagon, the octahedron, the four-dimensional cube and the 7-dimensional simplex.
%e From _Rajan Murthy_, Apr 08 2022: (Start)
%e 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.
%e For n = 12, there are a(12) = 4 regular polytopes with 12 hyperfaces. They, and their duals with 12 points, are:
%e 12 hyperfaces 12 points
%e dodecagon dodecagon
%e dodecahedron icosahedron
%e 6-cube 6-D orthoplex
%e 11-D simplex 11-D simplex
%e (End)
%o (PARI)
%o a(n)={
%o if(n<=3, return(1));
%o if(n==4||(n>=5&&n%2==1), return(2));
%o if(n>=6&&n%2==0, return(3+(n==12||n==20||n==24||n==120||n==600||(n>=8&&omega(2*n)==1))));
%o else(return(0));
%o } \\ _Jianing Song_, Sep 17 2018
%Y Cf. A302139 (indices of 4).
%K nonn,easy
%O 1,4
%A Paulo de A. Sachs (sachs6(AT)yahoo.de), Nov 09 2005
%E Terms beyond a(38) from _Jianing Song_, Sep 17 2018
%E a(1) and a(2) prepended and definition extended by _Rajan Murthy_, Apr 08 2022
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