The tables contain the

prime factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are suf ...

of the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

s 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 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 because the only ways ...

, the prime factorization is just ''n'' itself, written in bold below. The number 1 is called a

unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...

. It has no

prime factor A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only ways ...

s and is neither prime nor

composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ...

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 big Omega function, 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. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...

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 In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime nu ...

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 In number theory, a natural number is called ''k''-almost prime if it has ''k'' prime factors. More formally, a number ''n'' is ''k''-almost prime if and only if Ω(''n'') = ''k'', where Ω(''n'') is the total number of primes in the prime facto ...

(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 a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...

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 a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 ...

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 Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

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 In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

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 In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, ''n'' ...

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 A powerful number is a positive integer ''m'' such that for every prime number ''p'' dividing ''m'', ''p''2 also divides ''m''. Equivalently, a powerful number is the product of a square and a cube, that is, a number ''m'' of the form ''m'' = ''a ...

(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, 17 ...

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 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 factor of , is also a divisor. In other words, every prime factor appears at least squared in the factoriza ...

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-f ...

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 The Liouville Lambda function, denoted by λ(''n'') and named after Joseph Liouville, is an important arithmetic function. Its value is +1 if ''n'' is the product of an even number of prime numbers, and −1 if it is the product of an odd number of ...

λ(''n'') is 1 if Ω(''n'') is even, and is -1 if Ω(''n'') is odd. *The

Möbius function The Möbius function 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 most oft ...

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 In number theory, a sphenic number (from grc, σφήνα, 'wedge') is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers. Definit ...

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 the same prime only count 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 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 (n-1) \times (n-2) \t ...

''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 whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × ...

(for a natural number ''k'') has largest prime factor ≤ ''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 Regular numbers are numbers that evenly divide powers of 60 (or, equivalently, powers of 30). Equivalently, they are the numbers whose only prime divisors are 2, 3, and 5. As an example, 602 = 3600 = 48 ×&nb ...

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 In number theory, a frugal number is a natural number in a given number base that has more digits than the number of digits in its prime factorization in the given number base (including exponents). For example, in base 10, 125 = 53, 128 = 27, 243 ...

has more digits than the number of digits in its prime factorization (when written like below tables 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 of the Hindu–Arabic numeral ...

: 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 "Extravagant" is a song by American rapper Lil Durk recorded for his fourth studio album ''Love Songs 4 the Streets 2''. The song features vocals by Nicki Minaj. The song was produced by Julian Cruz and Skufl. Additional writing was done by Jose J ...

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) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

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 mathematics, 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 equivale ...

(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, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bot ...

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 divisible by ...

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

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See also