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[39] [40] The factorial number system is a mixed radix notation for numbers in which the place values of each digit are factorials. [ 41 ] Factorials are used extensively in probability theory , for instance in the Poisson distribution [ 42 ] and in the probabilities of random permutations . [ 43 ]
The factorial number system is sometimes defined with the 0! place omitted because it is always zero (sequence A007623 in the OEIS). In this article, a factorial number representation will be flagged by a subscript "!". In addition, some examples will have digits delimited by a colon. For example, 3:4:1:0:1:0! stands for
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
The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5×4×3×2×1 = 120. By convention, the value of 0! is defined as 1. This classical factorial function appears prominently in many theorems in number theory. The following are a few of these theorems. [1]
For example, the partitions of ! have lengths 4, 3 and 5. In other words, exactly one of the partitions of 5 ! {\textstyle 5!} has length 5. The number of sorted multiplicative partitions of n ! {\textstyle n!} that have length equal to n {\textstyle n} is 1 for n = 4 {\textstyle n=4} and n = 5 {\textstyle n=5} , and thereafter increases as
In number theory, the Kempner function [1] is defined for a given positive integer to be the smallest number such that divides the factorial!. For example, the number 8 {\displaystyle 8} does not divide 1 ! {\displaystyle 1!} , 2 ! {\displaystyle 2!} , or 3 ! {\displaystyle 3!} , but does divide 4 ! {\displaystyle 4!} , so S ( 8 ) = 4 ...
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The final expression is defined for all complex numbers except the negative even integers and satisfies (z + 2)!! = (z + 2) · z!! everywhere it is defined. As with the gamma function that extends the ordinary factorial function, this double factorial function is logarithmically convex in the sense of the Bohr–Mollerup theorem.
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