10,563 Pages

Torian
Notation$$T(n)$$
TypeCombinatorial
Based onFactorial
Growth rate$$f_{2}(n)$$
AuthorAalbert Torsius

The Torian is a function invented by Aalbert Torsius.[1] It is defined as $$T(x) = x!x$$, using Torsius' definition of the factorial:

$$x!n = \prod^{x}_{i = 1} i!(n - 1) = 1!(n - 1) \cdot 2!(n - 1) \cdot \ldots \cdot x!(n - 1)$$,

$$x!0 = x$$

This definition is a generalization of the ordinary factorial: $$x!1 = x!$$.

The first few values of $$x!x$$ for $$x = 0, 1, 2, 3, 4, 5, 6, 7, \cdots$$ are $$0, 1, 2, 24, 331\,776, 252\,428\,641\,478\,023\,533\,120, 1.8356962141506*10^{82}, 5.1012625185483*10^{315}, \cdots$$

## Faster method of computation

It is possible to calculate T(x) in a much faster way.

$$trn_x(n) = trn_{x-1}(1) + trn_{x-1}(2) + trn_{x-1}(3) \cdots trn_{x-1}(n)$$

$$trn_0(n) = n$$

The process of computing $$trn_x(n)$$ can be also shortened using formula $${n^{(x)} \over (n+1)!}$$, where $$n^{(x)}$$ is the rising factorial.

Here "$$trn_x$$" stands for order-x triangular number.

Then consider how it relates to x-order factorials:

$$n!2 = 2 \times (2 \times 3) \times (2 \times 3 \times 4) \ldots (2 \times 3 \times 4 \ldots (n-1) \times n)$$. Multiplication is commutative, and we know that 2 appears in that expression n-1 times, 3 appears n-2 times, and x appears n-(x-1) times.

Thus, $$n!2 = 2^{n-1} \times 3^{n-2} \times 4^{n-3} \times 5^{n-4} \cdots n$$ It can be also written as follows: $$n!2 = 2^{trn_0(n-1)} \times 3^{trn_0(n-2)} \times 4^{trn_0(n-3)} \times 5^{trn_0(n-4)} \cdots n$$

By the analogical considerations, $$n!3 = 2^{trn_1(n-1)} \times 3^{trn_1(n-2)} \times 4^{trn_1(n-3)} \times 5^{trn_1(n-4)} \cdots n$$

In general, $$n!x = 2^{trn_{x-2}(n-1)} \times 3^{trn_{x-2}(n-2)} \times 4^{trn_{x-2}(n-3)} \times 5^{trn_{x-2}(n-4)} \cdots n$$

## Pseudocode

// Torsius' factorial extension
function factorialTorsius(z, x):
if x = 0:
return z
result := 1
for i from 1 to z:
result := result * factorialTorsius(i, x - 1)
return result

// Torian
function torian(x):
return factorialTorsius(x, x)