Pernicious number

In number theory, a pernicious number is a positive integer where the Hamming weight (or digit sum) of its binary representation is prime.

Examples

The first pernicious number is 3, since 3 = 11₂ and 1 + 1 = 2, which is a prime. The next pernicious number is 5, since 5 = 101₂, followed by 6, 7 and 9 (sequence A052294 in the OEIS).

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 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 2^p−1(2^p − 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 2^p − 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).

Related numbers

• 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).

References

• The NumbersWithNames Program pp. 6–7.

External links

• Odious Numbers