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Encyclopedia > Composite number
 Divisibility-based sets of integers Form of factorization: Prime number Composite number Powerful number Square-free number Achilles number Constrained divisor sums: Perfect number Almost perfect number Quasiperfect number Multiply perfect number Hyperperfect number Superperfect number Unitary perfect number Semiperfect number Primitive semiperfect number Practical number Numbers with many divisors: Abundant number Highly abundant number Superabundant number Colossally abundant number Highly composite number Superior highly composite number Other: Deficient number Weird number Amicable number Friendly number Sociable number Solitary number Sublime number Harmonic divisor number Frugal number Equidigital number Extravagant number See also: Divisor function Divisor Prime factor Factorization This box: view • talk • edit

The first 15 composite numbers (sequence A002808 in OEIS) are The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, and 25.

• Every composite number can be written as the product of 2 or more (not necessarily distinct) primes (Fundamental theorem of arithmetic). ram
• Also, $(n-1)! ,,, equiv ,, 0 pmod{n}$ for all composite numbers n > 5. See also Wilson's theorem.

In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ... In mathematics, Wilsons theorem (also known as Al-Haythams theorem) states that p > 1 is a prime number if and only if (see factorial and modular arithmetic for the notation). ...

## Kinds of composite numbers

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 In mathematics, a semiprime (also called biprime or 2-almost prime, or pq number) is a natural number that is the product of two (not necessarily distinct) prime numbers. ... A sphenic number is a positive integer that is the product of three distinct prime factors. ...

$mu(n) = (-1)^{2x} = 1,$

(where μ is the Möbius function and x is half the total of prime factors), while for the former The classical MÃ¶bius function is an important multiplicative function in number theory and combinatorics. ...

$mu(n) = (-1)^{2x + 1} = -1.,$

Note however that for prime numbers the function also returns -1, and that μ(1) = 1. For a number n with one or more repeated prime factors, μ(n) = 0.

If all the prime factors of a number are repeated it is called a powerful number. If none of its prime factors are repeated, it is called squarefree. (All prime numbers and 1 are squarefree.) A powerful number is a positive integer m that for every prime number p dividing m, p2 also divides m. ... In mathematics, a square-free integer is one divisible by no perfect square, except 1. ...

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 {1,p,p2}. 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). A highly composite number is a positive integer which has more divisors than any positive integer below it. ...

Results from FactBites:

 PlanetMath: highly composite number (84 words) The first several superior highly composite numbers are 2, 6, 12, 60, 120, 360. "highly composite number" is owned by Kevin OBryant. This is version 4 of highly composite number, born on 2003-06-11, modified 2006-12-08.
 PlanetMath: composite number (120 words) A composite number is a positive integer which is not prime and not equal to 1. 2 is not composite, as it is prime. This is version 6 of composite number, born on 2002-05-23, modified 2004-09-10.
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