Pernicious number

Template:Short description 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.

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). The sequence of pernicious numbers begins Template:Bi

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. On the other hand, every number of the form $2^n+1$ with $n>1$, 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.

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

Related numbers

• Odious numbers are numbers with an odd number of 1s in their binary expansion (Template:OEIS2C).
• Evil numbers are numbers with an even number of 1s in their binary expansion (Template:OEIS2C).