Pernicious number

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In number theory, a pernicious number is a positive integer such that the Hamming weight of its binary representation is prime.


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).


  • 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 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 2p−1(2p − 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 2p − 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).

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