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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]
Factorion. In number theory, a factorion in a given number base is a natural number that equals the sum of the factorials of its digits. [1][2][3] The name factorion was coined by the author Clifford A. Pickover. [4]
Definition. 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)! (its place value). Radix/Base. 8.
Double factorial. The fifteen different chord diagrams on six points, or equivalently the fifteen different perfect matchings on a six-vertex complete graph. 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 ...
Stirling's approximation. Comparison of Stirling's approximation with the factorial. In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of .
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 (!
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