Innumber theory, apernicious number is a positive integer such that theHamming weight of itsbinary representation isprime, that is, there is a prime number of 1s when it is written as a binary number.^[1]

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

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, andone 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 everyFermat number, is a pernicious number. This is because the sum of the digits in binary form is 2, which is a prime number.^[2]

AMersenne number${\displaystyle 2^n-1}$ has a binary representation consisting of${\displaystyle n}$ ones, and is pernicious when${\displaystyle n}$ is prime. EveryMersenne prime is a Mersenne number for prime${\displaystyle n}$, and is therefore pernicious. By theEuclid–Euler theorem, the evenperfect 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]

• 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).

• Base-dependent integer sequences

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