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.[1]

Examples

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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

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, ... (sequence A052294 in the OEIS).

Properties

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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.[2] On the other hand, every number of the form   with  , 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.[2]

A Mersenne number   has a binary representation consisting of   ones, and is pernicious when   is prime. Every Mersenne prime is a Mersenne number for prime  , and is therefore pernicious. By the Euclid–Euler theorem, the even perfect numbers take the form   for a Mersenne prime  ; the binary representation of such a number consists of a prime number   of ones, followed by   zeros. Therefore, every even perfect number is pernicious.[3][4]

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References

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  1. ^ Deza, Elena (2021), Mersenne Numbers And Fermat Numbers, World Scientific, p. 263, ISBN 978-9811230332
  2. ^ a b c Sloane, N. J. A. (ed.), "Sequence A052294", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
  3. ^ Colton, Simon; Dennis, Louise (2002), "The NumbersWithNames Program", Seventh International Sumposium on Artificial Intelligence and Mathematics
  4. ^ Cai, Tianxin (2022), Perfect Numbers And Fibonacci Sequences, World Scientific, p. 50, ISBN 978-9811244094