Table of prime factors
The tables contain the prime factorization of the natural numbers from 1 to 1000.
When n is a prime number, 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 can be seen or directly computed from the prime factorization of.- The multiplicity of a prime factor of is the largest exponent for which divides. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is . The multiplicity of a prime which does not divide may be called or may be considered undefined.
- and, the prime omega functions, count the number of prime factors of a natural number.
- * is the number of distinct prime factors of.
- * is the number of prime factors of counted with multiplicity.
- A prime number has. 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 has. 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 . The first: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34.
- A -almost prime has .
- An even number has the prime factor 2. The first: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24.
- An odd number 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 has even multiplicity for all prime factors. The first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
- A cube has all multiplicities divisible by 3. The first: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728.
- A perfect power has a common divisor for all multiplicities. The first: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100. 1 is sometimes included.
- A powerful number has multiplicity greater than 1 for all its prime factors. The first: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72.
- A prime power has only one prime factor, i.e.. 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 has no prime factor with multiplicity greater than 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 greater than 1 is neither square-free nor squarefull, but squareful.
- The Liouville function is 1 if is even, and is -1 if is odd.
- The Möbius function is 0 if is not square-free. Otherwise is 1 if is even, and is −1 if is odd.
- A sphenic number is square-free and the product of 3 distinct primes, i.e. it has. The first: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154.
- , sometimes called the integer logarithm, is the sum of primes dividing, counted with multiplicity. It is an additive function.
- A Ruth-Aaron pair is a pair of two consecutive numbers with. The first : 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 : 5, 24, 49, 77, 104, 153, 369, 492, 714, 1682, 2107, 2299.
- A primorial is the product of all primes from 2 to. The first: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810. is sometimes included.
- A factorial is the product of all numbers from 1 to. The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600. is sometimes included.
- A -smooth number has its prime factors .
- is smoother than if the largest prime factor of is less than the largest of.
- A regular number has no prime factor greater than 5. The first: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16.
- A -powersmooth number has all where is a prime factor with multiplicity.
- A frugal number has more digits than the number of digits in its prime factorization. The first in decimal: 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.
- is the product of all prime factors which are both in and .
- and are coprime if they have no common prime factors, which implies.
- is the product of all prime factors of or .
- . Finding the prime factors is often harder than computing and using other algorithms which do not require known prime factorization.
- is a divisor of if all prime factors of have at least the same multiplicity in.
- The divisors of are all products of some or all prime factors of . The number of divisors can be computed by increasing all multiplicities by 1 and then multiplying them.