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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.
Every sequence of digits, in any base, is the sequence of initial digits of some factorial number in that base. [ 60 ] Another result on divisibility of factorials, Wilson's theorem , states that ( n − 1 ) ! + 1 {\displaystyle (n-1)!+1} is divisible by n {\displaystyle n} if and only if n {\displaystyle n} is a prime number . [ 52 ]
A general theory covering such relations, including the falling and rising factorial functions, is given by the theory of polynomial sequences of binomial type and Sheffer sequences. Falling and rising factorials are Sheffer sequences of binomial type, as shown by the relations: where the coefficients are the same as those in the binomial theorem.
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 . It is named after James Stirling, though a related but less precise result was first stated ...
In mathematics, the Fibonorial n!F, also called the Fibonacci factorial, where n is a nonnegative integer, is defined as the product of the first n positive Fibonacci numbers, i.e. where Fi is the ith Fibonacci number, and 0!F gives the empty product (defined as the multiplicative identity, i.e. 1).
In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in a purely algebraic setting in his book Methodus differentialis (1730). [1] They were rediscovered and given a combinatorial meaning by Masanobu Saka in 1782. [2]
Prefix sums are trivial to compute in sequential models of computation, by using the formula y i = y i − 1 + x i to compute each output value in sequence order. However, despite their ease of computation, prefix sums are a useful primitive in certain algorithms such as counting sort, [1] [2] and they form the basis of the scan higher-order function in functional programming languages.
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.