Prime signature

The prime signature of a number is the sequence of exponents of its prime factorisation sorted in order of size.

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 number of divisors that a number has is determined by its prime signature as follows : If you add one to each exponent and multiply them together you get the number of divisors including the number itself and 1. For example, 20 has prime signature {2,1} and so the number of divisors is 3x2=6. They are 1,2,4,5,10 and 20.

The smallest number of each prime signature is a product of primorials. The first few are:

:1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, ... OEIS|id=A025487.

Numbers with same prime signature

Sequences defined by their prime signature

Given a number with prime signature "S", it is
* A prime number if "S" = {1}
* A square if gcd "S" is even
* A square-free integer if max "S" = 1
* A powerful number if min "S" ≥ 2
* An Achilles number if min "S" ≥ 2 and gcd "S" = 1
* "k"-almost prime if sum "S" = "k"

References