Monday, April 15, 2013
MEMO: factorial ( n -- n! ) dup 1 > [ [1..b] product ] [ drop 1 ] if ;
Lots of Factorials
Similar to the factorial, the primorial is the product of the first n prime numbers.
The product of all the odd integers up to some odd positive integer n is called the double factorial of n, and denoted by n!!.
The multifactorial is a product of integers in steps of two (n!!, the “double factorial”), three (n!!!), or more (in general for a given k step: n!(k)).
The so-called quadruple factorial, however, is not the multifactorial n!(4); it is a much larger number given by (2n)!/n!.
The super factorial is the product of the first n factorials.
The hyper factorial is defined as:
The alternating factorial is the absolute value of the alternating sum of the first n factorials.
The exponential factorial is a positive integer n raised to the power of n−1, which in turn is raised to the power of n − 2, and so on and so forth.
Be careful with n > 4: the exponential factorial of 5 is 5262144 which is approximately 6.206069878660874 × 10183230.
A factorial prime is a prime number that is one less or one more than a factorial.
A primorial prime is a prime number that is one less or one more than a primorial.
The “descending factorial”, “falling sequential product”, or “lower factorial”.
A generalized version of falling factorial.
The “ascending factorial”, “rising sequential product”, or “upper factorial”.
That’s probably more factorials than anyone really cares to know about, but now Factor has more than ten times as many factorials as before! The code could probably use some cleanup, but it is available now in the math.factorials vocabulary.