Prime signature
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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. (A025487) The first few of which are
1,2,4,6,8,12,16,24,30,32,36,48,60,64, 72,96,120,128,144,180,192,210,216,...
