Compositorial

The compositorial, a portmanteau of composite and factorial, is formally defined as

\[c_n \#_c = \prod^{n}_{i = 1} c_i\]

where \(c_n\) is the nth composite number.

Another slightly more complex definition, which expands the domain of the function beyond composite numbers, is

\[n \#_c = \prod^{n-\pi (n)-1}_{i = 1} p_i\]

where \(c_n\) is the nth prime and \(\pi (n)\) is the prime counting function.

Using either definition, the compositorial of n can be informally defined as "the product of all composite numbers up to n, inclusive." For example, \(16 \#_c = 4 \cdot 6 \cdot 8 \cdot 9 \cdot 10 \cdot 12 \cdot 14 \cdot 15 \cdot 16 = 696,729,600\).

The sequence of compositorials goes:

1, 4, 24, 192, 1728, 17,280, 207,360, 2,903,040, ...