Factorial

Definition

Let $n$ be a nonnegative integer. The factorial of $n$, denoted $n!$, and read as n factorial, is defined as the product of all the natural numbers from $1$ to $n$. Note that $0!$ is defined as $1$.

$n!$ is also the order of the symmetric group, or the group of all permutations, on a set of size $n$.

Behavior

Initial values

The initial values of $k!$ for $k=0,1,2,3,4,\dots$ are given as:

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, [SHOW MORE]

3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000

View list on OEIS

Related notions

• lcm of all numbers so far: This is the exponential of the second Chebyshev function and also equals the exponent of the symmetric group of degree $n$.
• Maximum product over additive partitions
• Landau's function is the maximum lcm over additive partitions.
• Primorial is the product of the first few primes.