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

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

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

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

\[n! / n \# = \prod^{n-\pi (n)-1}_{i = 1} c_i\]

where \(c_n\) is the nth composite 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, \(16! / 16 \# = 4 \cdot 6 \cdot 8 \cdot 9 \cdot 10 \cdot 12 \cdot 14 \cdot 15 \cdot 16 = 696,729,600\).

The sequence of compositorials goes:

1, 4, 24, 192, 1,728, 17,280, 207,360, 2,903,040, ... (OEIS A036691)

Specific numbers

The number 192 is the compositorial of 8, and also the exponent in the tresexagintillion, which is the smallest positive integer whose full decimal expansion with thousands separators cannot be used at all as a page title in MediaWiki projects.

See also

Main article: Factorial
Multifactorials: Double factorial · Multifactorial
Falling and rising: Falling factorial · Rising factorial
Other mathematical variants: Alternating factorial · Hyperfactorial · q-factorial · Roman factorial · Subfactorial · Weak factorial · Primorial · Compositorial · Semiprimorial
Tetrational growth: Exponential factorial · Expostfacto function · Superfactorial by Clifford Pickover
Nested Factorials: Tetorial · Petorial · Ectorial · Zettorial · Yottorial
Array-based extensions: Hyperfactorial array notation · Nested factorial notation
Other googological variants: · Tetrofactorial · Superfactorial by Sloane and Plouffe · Torian · Factorexation · Mixed factorial · Bouncing Factorial
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