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.

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

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.
 * Evil numbers are numbers with an even number of 1s in their binary expansion.