Table of Prime Factors - Properties

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 pm divides n. The tables show the multiplicity for each prime factor. If no exponent is written then the multiplicity is 1 (since p = p1). 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 prime number has Ω(n) = 1. The first: 2, 3, 5, 7, 11 (sequence A000040 in OEIS). There are many special types of prime numbers.
• A composite number has Ω(n) > 1. The first: 4, 6, 8, 9, 10 (sequence A002808 in OEIS). 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 (sequence A001358 in OEIS).
• 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 (sequence A005843 in OEIS).
• An odd number does not have the prime factor 2. The first: 1, 3, 5, 7, 9 (sequence A005408 in OEIS). All integers are either even or odd.
• A square has even multiplicity for all prime factors (it is of the form a2 for some a). The first: 1, 4, 9, 16, 25 (sequence A000290 in OEIS).
• A cube has all multiplicities divisible by 3 (it is of the form a3 for some a). The first: 1, 8, 27, 64, 125 (sequence A000578 in OEIS).
• A perfect power has a common divisor m > 1 for all multiplicities (it is of the form am for some a > 1 and m > 1). The first: 4, 8, 9, 16, 25 (sequence A001597 in OEIS). 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 (sequence A001694 in OEIS).
• An Achilles number is powerful but not a perfect power. The first: 72, 108, 200, 288, 392 (sequence A052486 in OEIS).
• A square-free integer has no prime factor with multiplicity above 1. The first: 1, 2, 3, 5, 6 (sequence A005117 in OEIS)). 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 (sequence A007304 in OEIS).
• a₀(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 a₀(x) = a₀(x+1). The first (by x value): 5, 8, 15, 77, 125 (sequence A039752 in OEIS).
• A primorial x# is the product of all primes from 2 to x. The first: 2, 6, 30, 210, 2310 (sequence A002110 in OEIS). 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 (sequence A000142 in OEIS).
• A k-smooth number (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 has no prime factor above 5 (so it is 5-smooth). The first: 1, 2, 3, 4, 5, 6, 8 (sequence A051037 in OEIS).
• 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 (when written like below tables with multiplicities above 1 as exponents). The first in decimal: 125, 128, 243, 256, 343 (sequence A046759 in OEIS).
• An equidigital number has the same number of digits as its prime factorization. The first in decimal: 1, 2, 3, 5, 7, 10 (sequence A046758 in OEIS).
• An extravagant number has fewer digits than its prime factorization. The first in decimal: 4, 6, 8, 9, 12 (sequence A046760 in OEIS).
• 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 to compute gcd and lcm with 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.