Many properties of a natural numbern can be seen or directly computed from the prime factorization of n.
The multiplicity of a prime factorp of n is the largest exponent m for which pm divides n. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is 1. The multiplicity of a prime which does not divide n may be called 0 or may be considered undefined.
Ω, the big Omega function, is the number of prime factors of n counted with multiplicity.
A prime number has Ω = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37. There are many specialtypes of prime numbers.
A composite number has Ω > 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 = 2. The first: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34.
A k-almost prime has Ω = k.
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 divisorm > 1 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 above 1 for all prime factors. The first: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72.
A prime power 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 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.
A Ruth-Aaron pair is two consecutive numbers with a0 = a0. The first : 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 : 5, 24, 49, 77, 104, 153, 369, 492, 714, 1682, 2107, 2299
A primorialx# 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 factorialx! 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-powersmooth number has all pm ≤ 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. 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.
gcd is the product of all prime factors which are both in m and n.
lcm is the product of all prime factors of m or n.
gcd × lcm = 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 of n 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. 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.
