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. The first few are (sequence A025487 in OEIS):
- 1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, ...
[edit] Numbers with same prime signature
| Signature | Numbers | OEIS ID | Description |
|---|---|---|---|
| {1} | 2, 3, 5, 7, 11, ... | A000040 | prime numbers |
| {2} | 4, 9, 25, 49, 121, ... | A001248 | squares of prime numbers |
| {1,1} | 6, 10, 14, 15, 21, ... | A006881 | two distinct prime divisors (square-free semiprimes) |
| {3} | 8, 27, 125, 343, ... | A030078 | cubes of prime numbers |
| {2,1} | 12, 18, 20, 28, ... | A054753 | squares of primes times another prime |
| {4} | 16, 81, 625, 2401, ... | A030514 | fourth powers of prime numbers |
| {3,1} | 24, 40, 54, 56, ... | A065036 | cubes of primes times another prime |
| {1,1,1} | 30, 42, 66, 70, ... | A007304 | three distinct prime divisors (sphenic numbers) |
| {5} | 32, 243, 3125, ... | A050997 | fifth powers of primes |
| {2,2} | 36, 100, 196, 225, ... | A085986 | squares of square-free semiprimes |
