The prime signature of a positive integer is a sorted list of nonzero exponents in the prime factorization
By definition, the prime signature of 1 is . The prime numbers have prime signature and squares of prime numbers have prime signature .
The following table gives the prime signatures of the first few positive integers (OEIS A118914).
factorization | prime signature | factorization | prime signature | ||
1 | 1 | 11 | 11 | ||
2 | 2 | 12 | |||
3 | 3 | 13 | 13 | ||
4 | 14 | ||||
5 | 5 | 15 | |||
6 | 16 | ||||
7 | 7 | 17 | 17 | ||
8 | 18 | ||||
9 | 19 | 19 | |||
10 | 20 |