Editing Composite number (section)

==Types==

One way to classify composite numbers is by counting the number of prime factors. A composite number with two prime factors is a [[semiprime]] or 2-almost prime (the factors need not be distinct, hence squares of primes are included). A composite number with three distinct prime factors is a [[sphenic number]]. In some applications, it is necessary to differentiate between composite numbers with an odd number of distinct prime factors and those with an even number of distinct prime factors. For the latter 
:<math>\mu(n) = (-1)^{2x} = 1</math>

(where μ is the [[Möbius function]] and ''x'' is half the total of prime factors), while for the former

:<math>\mu(n) = (-1)^{2x + 1} = -1.</math>

However, for prime numbers, the function also returns −1 and <math>\mu(1) = 1</math>. For a number ''n'' with one or more repeated prime factors,

:<math>\mu(n) = 0</math>.{{sfn|Long|1972|p=159}}

If ''all'' the prime factors of a number are repeated it is called a [[powerful number]] (All [[perfect power]]s are powerful numbers). If ''none'' of its prime factors are repeated, it is called [[Square-free integer|squarefree]]. (All prime numbers and 1 are squarefree.)

For example, [[72 (number)|72]] = 2<sup>3</sup> × 3<sup>2</sup>, all the prime factors are repeated, so 72 is a powerful number. [[42 (number)|42]] = 2 × 3 × 7, none of the prime factors are repeated, so 42 is squarefree.

{{Euler_diagram_numbers_with_many_divisors.svg}}
Another way to classify composite numbers is by counting the number of divisors. All composite numbers have at least three divisors. In the case of squares of primes, those divisors are <math>\{1, p, p^2\}</math>. A number ''n'' that has more divisors than any ''x'' < ''n'' is a [[highly composite number]] (though the first two such numbers are 1 and 2).

Composite numbers have also been called "rectangular numbers", but that name can also refer to the [[pronic number]]s, numbers that are the product of two consecutive integers.

Yet another way to classify composite numbers is to determine whether all prime factors are either all below or all above some fixed (prime) number. Such numbers are called [[smooth number]]s and [[rough number]]s, respectively.