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12: 479 001 600: 13: 6 227 020 800: 14: 87 178 291 200: 15: ... the factorial of a non-negative integer ... , is the product of all positive integers less than or ...
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)!
These are counted by the double factorial 15 = (6 − 1)‼. In mathematics , the double factorial of a number n , denoted by n ‼ , is the product of all the positive integers up to n that have the same parity (odd or even) as n . [ 1 ]
The n-compositorial is equal to the n-factorial divided by the primorial n#. The compositorials are The compositorials are 1 , 4 , 24 , 192 , 1728 , 17 280 , 207 360 , 2 903 040 , 43 545 600 , 696 729 600 , ...
Let be a natural number. For a base >, we define the sum of the factorials of the digits [5] [6] of , :, to be the following: = =!. where = ⌊ ⌋ + is the number of digits in the number in base , ! is the factorial of and
In this article, the symbol () is used to represent the falling factorial, and the symbol () is used for the rising factorial. These conventions are used in combinatorics , [ 4 ] although Knuth 's underline and overline notations x n _ {\displaystyle x^{\underline {n}}} and x n ¯ {\displaystyle x^{\overline {n}}} are increasingly popular.
Since ! is the product of the integers 1 through n, we obtain at least one factor of p in ! for each multiple of p in {,, …,}, of which there are ⌊ ⌋.Each multiple of contributes an additional factor of p, each multiple of contributes yet another factor of p, etc. Adding up the number of these factors gives the infinite sum for (!
d() is the number of positive divisors of n, including 1 and n itself; σ() is the sum of the positive divisors of n, including 1 and n itselfs() is the sum of the proper divisors of n, including 1 but not n itself; that is, s(n) = σ(n) − n