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Pernicious number

Number with prime Hamming weight

Summary

Number with prime Hamming weight

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 n1, 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.

References

References

  1. Deza, Elena. (2021). "Mersenne Numbers And Fermat Numbers". World Scientific.
  2. {{cite OEIS. A052294
  3. (2002). "Seventh International Symposium on Artificial Intelligence and Mathematics".
  4. Cai, Tianxin. (2022). "Perfect Numbers And Fibonacci Sequences". World Scientific.
Wikipedia Source

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