# Sequence: total number of vertices in all n-dimensional regular polytopes

Jonathan Post jvospost3 at gmail.com
Mon Jul 14 21:57:31 CEST 2008

```Andrew's corrections are gratefully accepted.

I think that you should have your own sequence, which includes the
number of vertices of beautiful star polygons, star polyhedra and star
polytopes.

So, revised title and formula:

Total number of vertices in all finite n-dimensional convex regular
polytopes, or -1 if the number is infinite.

1, 2, -1, 106, 2453, 48, 83, 150, 281, 540, ...

For n>4, a(n) = A086653(n) + 1 =  2^n + 3*n + 1.

This is prime for n = 4, 6, 8, 12, 14, 22, 24, 46, 52, 54, 58, 62, ...

Example:
a(46) = (2^46) + 3*46 + 1 = 70368744177803 is prime.
a(52) = (2^52) + 3*52 + 1 = 4503599627370653 is prime.
a(54) = (2^54) + 3*54 + 1 = 18014398509482147 is prime.
a(58) = (2^58) + 3*58 + 1 = 288230376151711919 is prime.
a(62) = (2^62) + 3*62 + 1 = 4611686018427388091 is prime.
No more through a(100) = (2100) + 3×100 + 1 :

1267650600228229401496703205677 = 455008681 * 2785992120946433990117.

```