Pernicious number

In number theory, a pernicious number is a positive integer such that the Hamming weight of its binary representation is prime, that is, there is a prime number of 1s when it is written as a binary number.^[1]

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 (110₂), 7 (111₂) and 9 (1001₂).^[2] The sequence of pernicious numbers begins

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.^[2] On the other hand, every number of the form ${\displaystyle 2^{n}+1}$ with ${\displaystyle n>1}$, 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.^[2]

A Mersenne number ${\displaystyle 2^{n}-1}$ has a binary representation consisting of ${\displaystyle n}$ ones, and is pernicious when ${\displaystyle n}$ is prime. Every Mersenne prime is a Mersenne number for prime ${\displaystyle n}$, and is therefore pernicious. By the Euclid–Euler theorem, the even perfect numbers take the form ${\displaystyle 2^{n-1}(2^{n}-1)}$ for a Mersenne prime ${\displaystyle 2^{n}-1}$; the binary representation of such a number consists of a prime number ${\displaystyle n}$ of ones, followed by ${\displaystyle n-1}$ zeros. Therefore, every even perfect number is pernicious.^[3][4]

Related numbers

• Odious numbers are numbers with an odd number of 1s in their binary expansion (OEIS: A000069).
• Evil numbers are numbers with an even number of 1s in their binary expansion (OEIS: A001969).

References