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 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.
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 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.
If numbers are arranged in increasing columns of n numbers, then the prime factors of n will occur in the same row each time. The table columns have 20 = 22·5 numbers, so the prime factors 2 and 5 occur in fixed rows.