In mathematics, the prime signature of a number is the multiset of (nonzero) exponents of its prime factorization.
The prime signature of a number having prime factorization
For example, all prime numbers have a prime signature of {1}, the squares of primes have a prime signature of {2}, the products of 2 distinct primes have a prime signature of {1, 1} and the products of a square of a prime and a different prime (e.g. 12, 18, 20, ...) have a prime signature of {2, 1}.
The divisor function τ(n), the Möbius function μ(n), the number of distinct prime divisors ω(n) of n, the number of prime divisors Ω(n) of n, the indicator function of the squarefree integers, and many other important functions in number theory, are functions of the prime signature of n. In particular, τ(n) equals the product of the incremented by 1 exponents from the prime signature of n. For example, 20 has prime signature {2,1} and so the number of divisors is (2+1) × (1+1) = 6.
The smallest number of each prime signature is a product of primorials.
The first few are: A number cannot divide another unless its prime signature is included in the other numbers prime signature in the Young's lattice.
Given a number with prime signature S, it is