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Information about 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
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


A composite number is a positive integer which has a positive divisor other than one or itself. By definition, every integer greater than one is either a prime number or a composite number. The number one is considered to be neither prime nor composite. For example, the integer 14 is a composite number because it can be factored as 2 × 7.

The first 15 composite numbers (sequence [ A002808] in OEIS) are
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, and 25.

Properties

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



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



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

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 . 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).

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In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. An infinitude of prime numbers exists, as demonstrated by Euclid in about 300 BC.
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A powerful number is a positive integer m that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form
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In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32.
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An Achilles number is a number that is powerful but not a perfect power. A positive integer is a powerful number if, for every prime divisor or factor of , is also a divisor. In other words, every prime factor appears squared. All Achilles numbers are powerful.
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In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number itself.
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In mathematics, an almost perfect number (sometimes also called slightly defective number) is a natural number n such that the sum of all divisors of n (the divisor function σ(n)) is equal to 2n - 1.
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In mathematics, a quasiperfect number is a theoretical natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. Quasiperfect numbers are abundant numbers.
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In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number.

For a given natural number k, a number n is called k-perfect (or k
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In mathematics, a k-hyperperfect number (sometimes just called hyperperfect number) is a natural number n for which the equality n = 1 + k(σ(n) − n − 1) holds, where σ(
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A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. (A divisor d of a number n is a unitary divisor if d and n/d share no common factors.
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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 divisors.

The first few semiperfect numbers are
6, 12, 18, 20, 24, 28, 30, 36, 40, ...

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In mathematics, a primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect number) is a semiperfect natural number that has no semiperfect proper divisor.
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A practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n.
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In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. Here σ(n) is the sum-of-divisors function: the sum of all positive divisors of n, including n
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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 number less than it.
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In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. Formally, a natural number n is called superabundant precisely when, for any m < n,



where
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In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a certain kind of natural number. Formally, a number n is colossally abundant if and only if there is an ε > 0 such that for all k > 1,


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A highly composite number (HCN) is a positive integer which has more divisors than any smaller positive integer. (There is a second use of the term; see the section below.)

The first twenty-one highly composite numbers are listed in the table at right.
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In mathematics, a superior highly composite number is a certain kind of natural number. Formally, a natural number n is called superior highly composite iff there is an ε > 0 such that for all natural numbers k ≥ 1,


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In mathematics, a deficient number or defective number is a number n for which σ(n) < 2n. Here σ(n) is the sum-of-divisors function: the sum of all positive divisors of n
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weird number is a natural number that is abundant but not semiperfect. [1] In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.
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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 as a proper divisor but not the number itself.
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A friendly number is a positive natural number that shares a certain characteristic, to be defined below, with one or more other numbers. Two numbers sharing the property form a friendly pair. Larger clubs of mutually friendly numbers also exist.
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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 a sequence of numbers each of whose numbers is the sum of the factors of the preceding number, excluding the preceding
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A friendly number is a positive natural number that shares a certain characteristic, to be defined below, with one or more other numbers. Two numbers sharing the property form a friendly pair. Larger clubs of mutually friendly numbers also exist.
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In mathematics, a sublime number is a positive integer which has a perfect number of positive divisors (including itself), and whose positive divisors add up to another perfect number.[1]

The number 12, for example, is a sublime number.
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harmonic divisor number, or Ore number (named after Øystein Ore who defined it in 1948), is a positive integer whose divisors have a harmonic mean that is an integer.
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A frugal number is a natural number that has more digits than the number of digits in its prime factorization (including exponents). For example, using base-10 arithmetic, the first few frugal numbers are 125 (53), 128 (27), 243 (35
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An equidigital number is a number that has the same number of digits as the number of digits in its prime factorization (including exponents). For example, in base-10 arithmetic 1, 2, 3, 5, 7, and 10 (2×5) are equidigital numbers.
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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).
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