The semiprimorial, a portmanteau of semiprime and factorial, is formally defined as

\[\prod^{n}_{i = 1} s_i\]

where \(s_n\) is the nth semiprime number.

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

\[\prod^{sp(n)}_{i = 1} s_i\]

where \(c_n\) is the nth composite and \(sp(n)\) is the semiprime counting function.

Using either definition, the semiprimorial of n can be informally defined as "the product of all semiprime numbers up to n, inclusive." For example, the semiprimorial of 16 is equal to \(4 \cdot 6 \cdot 9 \cdot 10 \cdot 14 \cdot 15 = 453,600\).

The sequence of semiprimorials goes:

1, 4, 24, 216, 2,160, 30,240, 453,600, 9,525,600, ... (OEIS A112141)

Sources


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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