Pernicious number

Examples

Properties

• 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 to be a prime.
• Every number of the form with, 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 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 with prime p is a pernicious number known as a Mersenne number.

  Related numbers

• Odious numbers are numbers with an odd number of 1s in their binary expansion.
• Evil numbers are numbers with an even number of 1s in their binary expansion.