Divisibility-basedsets 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

A composite number is a positive integer which has a positive divisor other than one or itself. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 A powerful number is a Positive integer m that for every prime number p dividing m, p 2 also divides m In Mathematics, a square-free, or quadratfrei, Integer is one divisible by no perfect square, except 1 An Achilles number is a number that is powerful but not a Perfect power. In mathematics a perfect number is defined as a positive integer which is the sum of its proper positive Divisors that is the sum of the positive divisors excluding In Mathematics, an almost perfect number (sometimes also called slightly defective number) is a Natural number n such that the Sum In Mathematics, a quasiperfect number is a theoretical Natural number n for which the sum of all its Divisors (the Divisor function In Mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a Perfect number. In Mathematics, a k -hyperperfect number (sometimes just called hyperperfect number) is a Natural number n for which the equality In mathematics a superperfect number is a positive Integer n that satisfies \sigma^2(n=\sigma(\sigma(n=2n\, where σ is A unitary perfect number is an Integer which is the sum of its positive proper Unitary divisors not including the number itself In Mathematics, a semiperfect number or pseudoperfect number is a Natural number n that is equal to the sum of all or some of its Proper In Mathematics, a primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect In Mathematics, and in particular Number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive In Mathematics, an abundant number or excessive number is a number n for which σ ( n) > 2 n. In Mathematics, a highly abundant number is a Natural number where the sum of its divisors (including itself is greater than the sum of the divisors of any natural In Mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of Natural number. In Mathematics, a colossally abundant number (sometimes abbreviated as CA) is a certain kind of Natural number. A highly composite number ( HCN) is a positive Integer with more Divisors than any smaller positive integer In Mathematics, a superior highly composite number is a certain kind of Natural number. In Mathematics, a deficient number or defective number is a number n for which σ ( n)  n. In Mathematics, a weird number is a Natural number that is abundant but not semiperfect. Amicable numbers are two different Numbers so related that the sum of the Proper divisors of the one is equal to the other one being considered In Number theory, a friendly number is a Natural number that shares a certain characteristic called abundancy, the ratio between the sum of Divisors Sociable numbers are generalizations of the concepts of Amicable numbers and Perfect numbers A set of sociable numbers is a kind of Aliquot sequence, or In Number theory, a friendly number is a Natural number that shares a certain characteristic called abundancy, the ratio between the sum of Divisors In mathematics a sublime number is a positive Integer which has a Perfect number of positive Divisors (including itself and whose positive divisors add In Mathematics, a harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948) is a positive integer whose divisors A frugal number is a natural number that has more digits than the number of digits in its prime factorization (including exponents An equidigital number is a number that has the same number of digits as the number of digits in its prime factorization (including exponents An extravagant number (also known as a wasteful number is a natural number that has fewer digits than the number of digits in its prime factorization (including exponents In Mathematics, and specifically in Number theory, a divisor function is an Arithmetical function related to the Divisors of an Integer In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without In Number theory, the prime factors of a positive Integer are the Prime numbers that divide into that integer exactly without leaving a remainder In Mathematics, factorization ( also factorisation in British English) or factoring is the decomposition of an object (for A negative number is a Number that is less than zero, such as −2 The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a divisor of an Integer n, also called a factor of n, is an integer which evenly divides n without In other words, if 0 < n is an integer and there are integers 1 < a, b < n such that n = a × b then n is composite. By definition, every integer greater than one is either a prime number or a composite number. Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 The number one is a unit - it is neither prime nor composite. Mathematics For any number x: x ·1 = 1· x = x (1 is the multiplicative identity In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i For example, the integer 14 is a composite number because it can be factored as 2 × 7.

The first 89 composite numbers (sequence A002808 in OEIS) are

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120. The On-Line Encyclopedia of Integer Sequences ( OEIS) also cited simply as Sloane's, is an extensive searchable Database of Integer sequences

• Every composite number can be written as the product of 2 or more (not necessarily distinct) primes (Fundamental theorem of arithmetic). In Number theory, the fundamental theorem of arithmetic (or unique-prime-factorization theorem) states that every Natural number greater than 1 can be written
• Also, $(n-1)! \,\,\, \equiv \,\, 0 \pmod{n}$ for all composite numbers n > 5. See also Wilson's theorem. In Mathematics, Wilson's theorem states that p > 1 is a Prime number If and only if (p-1!\ \equiv\ -1\ (\mbox{mod}\ p

## 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). In Mathematics, a semiprime (also called biprime or 2- Almost prime, or pq number) is a Natural number that is the product A composite number with three distinct prime factors is a sphenic number. In Mathematics, a sphenic number ( Old Greek sphen = Wedge) is a positive integer which is the product of three distinct Prime 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

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

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

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

Note however that for prime numbers the function also returns -1, and that μ(1) = 1. For the rational functions defined on the complex numbers see Möbius transformation. 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. A powerful number is a Positive integer m that for every prime number p dividing m, p 2 also divides m If none of its prime factors are repeated, it is called squarefree. In Mathematics, a square-free, or quadratfrei, Integer is one divisible by no perfect square, except 1 (All prime numbers and 1 are squarefree. )

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 ( HCN) is a positive Integer with more Divisors than any smaller positive integer