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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
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A composite number is a positive integer which has a positive divisor other than one or itself. 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. The number one is a unit - it is neither prime nor composite. For example, the integer 14 is a composite number because it can be factored as 2 × 7. In mathematics, a prime number (or a prime) is a natural number greater than 1 which has exactly two distinct natural number divisors: 1 and itself. ... 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, 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 an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... 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. ... In mathematics, a quasiperfect number is a natural number n for which the sum of all its divisors (the divisor function σ(n)) is equal to 2n + 1. ... 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 n = 1 + k(σ(n) − n − 1) holds, where σ(n) is the divisor function (i. ... 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 divisors. ... In mathematics, a primitive semiperfect number (also called a primitive pseudoperfect number, irreducible semiperfect number or irreducible pseudoperfect number) is a natural number that has no semiperfect proper divisor. ... A practical number or panarithmic number is a positive integer n such that all preceding positive integers are a sum of distinct divisors of n. ... In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. ... In mathematics, a highly abundant number is a certain kind of natural number. ... 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 is a positive integer which has more divisors than any positive integer below it. ... 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) < 2n. ... The term weird number also refers to a phenomenon in twos complement arithmetic. ... Amicable numbers are two numbers so related that the sum of the proper divisors of the one is equal to the other, unity being considered as a proper divisor but not the number itself. ... A friendly number is a positive natural number that shares a certain characteristic, to be defined below, with one or more other numbers. ... Sociable numbers are generalizations of the concepts of amicable numbers and perfect numbers. ... In mathematics a solitary number is number which does not have any friends. Two numbers m and n are friends if and only if σ(m)/m = σ(n)/n. ... 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. ... A harmonic divisor number, or Ore number, is a number whose divisors, averaged in a harmonic mean, results in an integer. ... 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). ... Divisor function σ0(n) up to n=250 Sigma function σ1(n) up to n=250 Sum of the squares of divisors, σ2(n), up to n=250 Sum of cubes of divisors, σ3(n) up to n=250 In mathematics, and specifically in number theory, a divisor function is... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... This article is about the concept in number theory. ... ... A negative number is a number that is less than zero, such as −3. ... Not to be confused with Natural number. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... This article is about the number one. ... In mathematics, a prime number (or a prime) is a natural number greater than 1 which has exactly two distinct natural number divisors: 1 and itself. ... This article is about the number one. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of...


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.

Properties

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 Möbius 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. ...


External links


  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.
  More results at FactBites »

 
 

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