Editing Table of prime factors (section)

== 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<sup>m</sup>'' divides ''n''. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is 1 (since ''p'' = ''p''<sup>1</sup>). 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]], is the number of prime factors of ''n'' counted with multiplicity (so it is the sum of all prime factor multiplicities).
*A [[prime number]] has Ω(''n'') = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 {{OEIS|id=A000040}}. There are many special [[List of prime numbers|types of prime numbers]].
*A [[composite number]] has Ω(''n'') > 1. The first: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21 {{OEIS|id=A002808}}. 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 {{OEIS|id=A001358}}.
*A ''k''-[[almost prime]] (for a natural number ''k'') has Ω(''n'') = ''k'' (so it is composite if ''k'' > 1).
*An [[even number]] has the prime factor 2. The first: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 {{OEIS|id=A005843}}.
*An [[odd number]] does not have the prime factor 2. The first: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 {{OEIS|id=A005408}}. All integers are either even or odd.
*A [[Square number|square]] has even multiplicity for all prime factors (it is of the form ''a''<sup>2</sup> for some ''a''). The first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 {{OEIS|id=A000290}}.
*A [[Cube (algebra)|cube]] has all multiplicities divisible by 3 (it is of the form ''a''<sup>3</sup> for some ''a''). The first: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 {{OEIS|id=A000578}}.
*A [[perfect power]] has a common divisor ''m'' > 1 for all multiplicities (it is of the form ''a<sup>m</sup>'' for some ''a'' > 1 and ''m'' > 1). The first: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100 {{OEIS|id=A001597}}. 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 {{OEIS|id=A001694}}.
*A [[prime power]] has only one prime factor. The first: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19 {{OEIS|id=A000961}}. 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 {{OEIS|id=A052486}}.
*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 {{OEIS|id=A005117}}. 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]] 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 {{OEIS|id=A007304}}.
*''a''<sub>0</sub>(''n'') is the sum of primes dividing ''n'', counted with multiplicity. It is an [[additive function]].
*A [[Ruth-Aaron pair]] is two consecutive numbers (''x'', ''x''+1) with ''a''<sub>0</sub>(''x'') = ''a''<sub>0</sub>(''x''+1). The first (by ''x'' value): 5, 8, 15, 77, 125, 714, 948, 1330, 1520, 1862, 2491, 3248 {{OEIS|id=A039752}}. 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 {{OEIS|id=A006145}}.
*A [[primorial]] ''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 {{OEIS|id=A002110}}. 1# = 1 is sometimes included.
*A [[factorial]] ''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 {{OEIS|id=A000142}}. 0! = 1 is sometimes included.
*A ''k''-[[smooth number]] (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 {{OEIS|id=A051037}}.
*A ''k''-[[Smooth number#Powersmooth numbers|powersmooth]] number has all ''p''<sup>''m''</sup> ≤ ''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]]: 125, 128, 243, 256, 343, 512, 625, 729, 1024, 1029, 1215, 1250 {{OEIS|id=A046759}}.
*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 {{OEIS|id=A046758}}.
*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 {{OEIS|id=A046760}}.
*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]] 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]] (also called relatively prime) if gcd(''m'', ''n'') = 1 (meaning they have no common prime factor).
*lcm(''m'', ''n'') ([[least common multiple]] 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]] 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]].