The tables contain the

prime factorization In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a comp ...

of the

natural numbers In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positiv ...

from 1 to 1000. When ''n'' is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

, the prime factorization is just ''n'' itself, written in bold below. The number 1 is called a unit. It has no prime factors and is neither prime nor composite.

Properties

Many properties of a natural number ''n'' can be seen or directly computed from the prime factorization of ''n''. *The multiplicity of a prime factor ''p'' of ''n'' is the largest exponent ''m'' for which ''p^m'' divides ''n''. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is 1 (since ''p'' = ''p''¹). The multiplicity of a prime which does not divide ''n'' may be called 0 or may be considered undefined. *Ω(''n''), the

prime omega function In number theory, the prime omega functions \omega(n) and \Omega(n) count the number of prime factors of a natural number n. The number of ''distinct'' prime factors is assigned to \omega(n) (little omega), while \Omega(n) (big omega) counts the '' ...

, is the number of prime factors of ''n'' counted with multiplicity (so it is the sum of all prime factor multiplicities). *A

has Ω(''n'') = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 . There are many special types of prime numbers. *A

composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Accordingly it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime numb ...

has Ω(''n'') > 1. The first: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21 . All numbers above 1 are either prime or composite. 1 is neither. *A semiprime has Ω(''n'') = 2 (so it is composite). The first: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34 . *A ''k''- almost prime (for a natural number ''k'') has Ω(''n'') = ''k'' (so it is composite if ''k'' > 1). *An

even number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The ...

has the prime factor 2. The first: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 . *An

odd number In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The ...

does not have the prime factor 2. The first: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 . All integers are either even or odd. *A

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

has even multiplicity for all prime factors (it is of the form ''a''² for some ''a''). The first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 . *A

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

has all multiplicities divisible by 3 (it is of the form ''a''³ for some ''a''). The first: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 . *A perfect power has a common divisor ''m'' > 1 for all multiplicities (it is of the form ''a^m'' for some ''a'' > 1 and ''m'' > 1). The first: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100 . 1 is sometimes included. *A powerful number (also called squareful) has multiplicity above 1 for all prime factors. The first: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72 . *A

prime power In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 1 ...

has only one prime factor. The first: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19 . 1 is sometimes included. *An Achilles number is powerful but not a perfect power. The first: 72, 108, 200, 288, 392, 432, 500, 648, 675, 800, 864, 968 . *A

square-free integer In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, is square-fr ...

has no prime factor with multiplicity above 1. The first: 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17 . A number where some but not all prime factors have multiplicity above 1 is neither square-free nor squareful. *The Liouville function λ(''n'') is 1 if Ω(''n'') is even, and is -1 if Ω(''n'') is odd. *The

Möbius function The Möbius function \mu(n) is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and m ...

μ(''n'') is 0 if ''n'' is not square-free. Otherwise μ(''n'') is 1 if Ω(''n'') is even, and is −1 if Ω(''n'') is odd. *A sphenic number has Ω(''n'') = 3 and is square-free (so it is the product of 3 distinct primes). The first: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154 . *''a''₀(''n'') is the sum of primes dividing ''n'', counted with multiplicity. It is an

additive function In number theory, an additive function is an arithmetic function ''f''(''n'') of the positive integer variable ''n'' such that whenever ''a'' and ''b'' are coprime, the function applied to the product ''ab'' is the sum of the values of the funct ...

. *A Ruth-Aaron pair is two consecutive numbers (''x'', ''x''+1) with ''a''₀(''x'') = ''a''₀(''x''+1). The first (by ''x'' value): 5, 8, 15, 77, 125, 714, 948, 1330, 1520, 1862, 2491, 3248 . Another definition is where the same prime is only counted once; if so, the first (by ''x'' value): 5, 24, 49, 77, 104, 153, 369, 492, 714, 1682, 2107, 2299 . *A

primorial In mathematics, and more particularly in number theory, primorial, denoted by "", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function ...

''x''# is the product of all primes from 2 to ''x''. The first: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810 . 1# = 1 is sometimes included. *A

factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...

''x''! is the product of all numbers from 1 to ''x''. The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 . 0! = 1 is sometimes included. *A ''k''-

smooth number In number theory, an ''n''-smooth (or ''n''-friable) number is an integer whose prime factors are all less than or equal to ''n''. For example, a 7-smooth number is a number in which every prime factor is at most 7. Therefore, 49 = 72 and 15750 = 2 ...

(for a natural number ''k'') has its prime factors ≤ ''k'' (so it is also ''j''-smooth for any ''j'' > ''k''). *''m'' is smoother than ''n'' if the largest prime factor of ''m'' is below the largest of ''n''. *A regular number has no prime factor above 5 (so it is 5-smooth). The first: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16 . *A ''k''- powersmooth number has all ''p''^''m'' ≤ ''k'' where ''p'' is a prime factor with multiplicity ''m''. *A frugal number has more digits than the number of digits in its prime factorization (when written like the tables below with multiplicities above 1 as exponents). The first in

decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...

: 125, 128, 243, 256, 343, 512, 625, 729, 1024, 1029, 1215, 1250 . *An equidigital number has the same number of digits as its prime factorization. The first in decimal: 1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17 . *An extravagant number has fewer digits than its prime factorization. The first in decimal: 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30 . *An economical number has been defined as a frugal number, but also as a number that is either frugal or equidigital. *gcd(''m'', ''n'') (

greatest common divisor In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest co ...

of ''m'' and ''n'') is the product of all prime factors which are both in ''m'' and ''n'' (with the smallest multiplicity for ''m'' and ''n''). *''m'' and ''n'' are

coprime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...

(also called relatively prime) if gcd(''m'', ''n'') = 1 (meaning they have no common prime factor). *lcm(''m'', ''n'') (

least common multiple In arithmetic and number theory, the least common multiple (LCM), lowest common multiple, or smallest common multiple (SCM) of two integers ''a'' and ''b'', usually denoted by , is the smallest positive integer that is divisible by both ''a'' and ...

of ''m'' and ''n'') is the product of all prime factors of ''m'' or ''n'' (with the largest multiplicity for ''m'' or ''n''). *gcd(''m'', ''n'') × lcm(''m'', ''n'') = ''m'' × ''n''. Finding the prime factors is often harder than computing gcd and lcm using other algorithms which do not require known prime factorization. *''m'' is a

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

of ''n'' (also called ''m'' divides ''n'', or ''n'' is divisible by ''m'') if all prime factors of ''m'' have at least the same multiplicity in ''n''. The divisors of ''n'' are all products of some or all prime factors of ''n'' (including the empty product 1 of no prime factors). The number of divisors can be computed by increasing all multiplicities by 1 and then multiplying them. Divisors and properties related to divisors are shown in table of divisors.

Properties

1 to 100

101 to 200

201 to 300

301 to 400

401 to 500

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601 to 700

701 to 800

801 to 900

901 to 1000

See also