From Wikipedia, the free encyclopedia
- No power of two is a pernicious number. This is trivially true, because powers of two in binary form are represented as a one followed by zeros. So each power of two has a Hamming weight of one, and one is not considered a prime.
- Every number of the form 2n + 1 with n>0, including every Fermat number, is a pernicious number. This is because the sum of the digits in binary form is 2, which is a prime number.
- Every even perfect number is a pernicious number. This is based on the fact that every even perfect number can be represented as 2p−1(2p − 1) with p a prime. Owing to this form, every even perfect number is represented in binary as p ones followed by p − 1 zeros.
- A number of the form 2p − 1 with prime p is a pernicious number known as a Mersenne number (although sometimes Mersenne numbers are defined as 2n − 1 for any natural number n).
- Odious numbers are numbers with an odd number of 1s in their binary expansion (sequence A000069 in the OEIS).
- Evil numbers are numbers with an even number of 1s in their binary expansion (sequence A001969 in the OEIS).
- The NumbersWithNames Program pp. 6–7.