Table of prime factors

Many properties of a natural number ${\displaystyle n}$ can be seen or directly computed from the prime factorization of ${\displaystyle n}$.

• The multiplicity of a prime factor ${\displaystyle p}$ of ${\displaystyle n}$ is the largest exponent ${\displaystyle m}$ for which ${\displaystyle p^{m}}$ divides ${\displaystyle n}$. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is ${\displaystyle 1}$ (since ${\displaystyle p=p^{1}}$). The multiplicity of a prime which does not divide ${\displaystyle n}$ may be called ${\displaystyle 0}$ or may be considered undefined.
• ${\displaystyle \omega (n)}$ and ${\displaystyle \Omega (n)}$, the prime omega functions, count the number of prime factors of a natural number ${\displaystyle n}$.
  • ${\displaystyle \omega (n)}$ (little omega) is the number of distinct prime factors of ${\displaystyle n}$.
  • ${\displaystyle \Omega (n)}$ (big omega) is the number of prime factors of ${\displaystyle n}$ counted with multiplicity (so it is the sum of all prime factor multiplicities).
• A prime number has ${\displaystyle \Omega (n)=\omega (n)=1}$. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 (sequence A000040 in the OEIS). There are many special types of prime numbers.
• A composite number has ${\displaystyle \Omega (n)\geq \omega (n)>1}$. The first: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21 (sequence A002808 in the OEIS). All numbers above 1 are either prime or composite. 1 is neither.
• A semiprime has ${\displaystyle \Omega (n)=2}$ (so it is composite). The first: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34 (sequence A001358 in the OEIS).
• A ${\displaystyle k}$-almost prime (for a natural number ${\displaystyle k}$) has ${\displaystyle \Omega (n)=k}$ (so it is composite if ${\displaystyle k>1}$).
• An even number has the prime factor 2. The first: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 (sequence A005843 in the OEIS).
• An odd number does not have the prime factor 2. The first: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 (sequence A005408 in the OEIS). All integers are either even or odd.
• A square has even multiplicity for all prime factors (it is of the form ${\displaystyle a^{2}}$ for some ${\displaystyle a}$). The first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 (sequence A000290 in the OEIS).
• A cube has all multiplicities divisible by 3 (it is of the form ${\displaystyle a^{3}}$ for some ${\displaystyle a}$). The first: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 (sequence A000578 in the OEIS).
• A perfect power has a common divisor ${\displaystyle m>1}$ for all multiplicities (it is of the form ${\displaystyle a^{m}}$ for some ${\displaystyle a>1}$ and ${\displaystyle m>1}$). The first: 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100 (sequence A001597 in the OEIS). 1 is sometimes included.
• A powerful number (also called squarefull) has multiplicity greater than 1 for all its prime factors. The first: 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72 (sequence A001694 in the OEIS).
• A prime power has only one prime factor, i.e. ${\displaystyle \omega (n)=1}$. The first: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19 (sequence A000961 in the OEIS). 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 (sequence A052486 in the OEIS).
• 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 (sequence A005117 in the OEIS). 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 ${\displaystyle \lambda (n)}$ is 1 if ${\displaystyle \Omega (n)}$ is even, and is -1 if ${\displaystyle \Omega (n)}$ is odd.
• The Möbius function ${\displaystyle \mu (n)}$ is 0 if ${\displaystyle n}$ is not square-free. Otherwise ${\displaystyle \mu (n)}$ is 1 if ${\displaystyle \Omega (n)}$ is even, and is −1 if ${\displaystyle \Omega (n)}$ is odd.
• A sphenic number is square-free and the product of 3 distinct primes, i.e. it has ${\displaystyle \omega (n)=\Omega (n)=3}$. The first: 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154 (sequence A007304 in the OEIS).
• ${\displaystyle a_{0}(n)}$, sometimes called the integer logarithm, is the sum of primes dividing ${\displaystyle n}$, counted with multiplicity. It is an additive function.
• A Ruth-Aaron pair is a pair of two consecutive numbers ${\displaystyle (n,n+1)}$ with ${\displaystyle a_{0}(n)=a_{0}(n+1)}$. The first (by ${\displaystyle n}$ value): 5, 8, 15, 77, 125, 714, 948, 1330, 1520, 1862, 2491, 3248 (sequence A039752 in the OEIS). Another definition is where the same prime is only counted once; if so, the first (by ${\displaystyle n}$ value): 5, 24, 49, 77, 104, 153, 369, 492, 714, 1682, 2107, 2299 (sequence A006145 in the OEIS).
• A primorial ${\displaystyle p_{n}\#}$ is the product of all primes from 2 to ${\displaystyle p_{n}}$. The first: 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810 (sequence A002110 in the OEIS). ${\displaystyle 1\#=1}$ is sometimes included.
• A factorial ${\displaystyle n!}$ is the product of all numbers from 1 to ${\displaystyle n}$. The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 (sequence A000142 in the OEIS). ${\displaystyle 0!=1}$ is sometimes included.
• A ${\displaystyle k}$-smooth number (for a natural number ${\displaystyle k}$) has its prime factors ${\displaystyle \leq k}$ (so it is also ${\displaystyle j}$-smooth for any ${\displaystyle j>k}$).
• ${\displaystyle m}$ is smoother than ${\displaystyle n}$ if the largest prime factor of ${\displaystyle m}$ is less than the largest of ${\displaystyle n}$.
• A regular number has no prime factor greater than 5 (so it is 5-smooth). The first: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16 (sequence A051037 in the OEIS).
• A ${\displaystyle k}$-powersmooth number has all ${\displaystyle p^{m}\leq k}$ where ${\displaystyle p}$ is a prime factor with multiplicity ${\displaystyle m}$.
• 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 (sequence A046759 in the OEIS).
• 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 (sequence A046758 in the OEIS).
• 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 (sequence A046760 in the OEIS).
• An economical number has been defined as a frugal number, but also as a number that is either frugal or equidigital.
• ${\displaystyle gcd(m,n)}$ (greatest common divisor of ${\displaystyle m}$ and ${\displaystyle n}$) is the product of all prime factors which are both in ${\displaystyle m}$ and ${\displaystyle n}$ (with the smallest multiplicity for ${\displaystyle m}$ and ${\displaystyle n}$).
• ${\displaystyle m}$ and ${\displaystyle n}$ are coprime (also called relatively prime) if they have no common prime factors, which implies ${\displaystyle gcd(m,n)=1}$.
• ${\displaystyle lcm(m,n)}$ (least common multiple of ${\displaystyle m}$ and ${\displaystyle n}$) is the product of all prime factors of ${\displaystyle m}$ or ${\displaystyle n}$ (with the largest multiplicity for ${\displaystyle m}$ or ${\displaystyle n}$).
• ${\displaystyle gcd(m,n)\times lcm(m,n)=m\times n}$. Finding the prime factors is often harder than computing ${\displaystyle gcd}$ and ${\displaystyle lcm}$ using other algorithms which do not require known prime factorization.
• ${\displaystyle m}$ is a divisor of ${\displaystyle n}$ (also called ${\displaystyle m}$ divides ${\displaystyle n}$, or ${\displaystyle n}$ is divisible by ${\displaystyle m}$) if all prime factors of ${\displaystyle m}$ have at least the same multiplicity in ${\displaystyle n}$.
• The divisors of ${\displaystyle n}$ are all products of some or all prime factors of ${\displaystyle n}$ (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.

• Fundamental theorem of arithmetic – Integers have unique prime factorizations
• List of prime numbers
• Table of divisors