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In mathematics, the factorial of a non-negative integer, denoted by !, is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial: ! = () = ()! For example, ! =! = =
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
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.
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
Peter Luschny, Approximation formulas for the factorial function n! Weisstein, Eric W. , "Stirling's Approximation" , MathWorld Stirling's approximation at PlanetMath .
Multiplicative partitions of factorials are expressions of values of the factorial function as products of powers of prime numbers. They have been studied by Paul Erdős and others. [1] [2] [3] The factorial of a positive integer is a product of decreasing integer factors, which can in turn be factored into prime numbers.
The number 9! is the lowest factorial which is multiple of 810, so the proper factor 811 is found in this step. The factor 139 is not found this time because p−1 ...
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]