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Use divide and conquer to compute the product of the primes whose exponents are odd; Divide all of the exponents by two (rounding down to an integer), recursively compute the product of the prime powers with these smaller exponents, and square the result; Multiply together the results of the two previous steps
The n-compositorial is equal to the n-factorial divided by the primorial n#. The compositorials are 1, 4, 24, 192, 1728, 17 280, 207 360, 2 903 040, 43 545 600, 696 ...
The factorial number system is a mixed radix numeral system: the i-th digit from the right has base i, which means that the digit must be strictly less than i, and that (taking into account the bases of the less significant digits) its value is to be multiplied by (i − 1)!
I propose to write !! for such products, and if a name be required for the product to call it the "alternate factorial" or the "double factorial". Meserve (1948) [9] states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals that arise in the derivation of the Wallis product.
As one special case, it can be used to prove that if n is a positive integer then 4 divides () if and only if n is not a power of 2. It follows from Legendre's formula that the p -adic exponential function has radius of convergence p − 1 / ( p − 1 ) {\displaystyle p^{-1/(p-1)}} .
The multiplicity of a prime which does not divide n may be called 0 or may be ... The first: 4, 6, 9, 10, 14, 15, 21 ... A factorial x! is the product of all ...
For each of the values of n from 2 to 30, the following table shows the number (n − 1)! and the remainder when (n − 1)! is divided by n. (In the notation of modular arithmetic, the remainder when m is divided by n is written m mod n.) The background color is blue for prime values of n, gold for composite values.
Computing: 9.999 999 × 10 96 is equal to the largest value that can be represented in the IEEE decimal32 floating-point format. Computing: 69! (roughly 1.7112245 × 10 98), is the largest factorial value that can be represented on a calculator with two digits for powers of ten without overflow.