Pentatope number

It is named because it represents the number of 3-dimensional unit spheres which can be packed into a pentatope (a 4-dimensional tetrahedron) of increasing side lengths.

[1] The formula for the nth pentatope number is represented by the 4th rising factorial of n divided by the factorial of 4: The pentatope numbers can also be represented as binomial coefficients: which is the number of distinct quadruples that can be selected from n + 3 objects, and it is read aloud as "n plus three choose four".

[2] The infinite sum of the reciprocals of all pentatope numbers is ⁠4/3⁠.

Similarly, the only primes preceding a 6-simplex number are 83 and 461.

We can derive this test from the formula for the nth pentatope number.

Given a positive integer x, to test whether it is a pentatope number we can compute the positive root using Ferrari's method: The number x is pentatope if and only if n is a natural number.

In that case x is the nth pentatope number.

The generating function for pentatope numbers is[4] In biochemistry, the pentatope numbers represent the number of possible arrangements of n different polypeptide subunits in a tetrameric (tetrahedral) protein.