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.^[1]

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).^[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

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 ${\displaystyle 2^{n}+1}$ with ${\displaystyle n>1}$, 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.^[2]

A Mersenne number ${\displaystyle 2^{n}-1}$ has a binary representation consisting of ${\displaystyle n}$ ones, and is pernicious when ${\displaystyle n}$ is prime. Every Mersenne prime is a Mersenne number for prime ${\displaystyle n}$, and is therefore pernicious. By the Euclid–Euler theorem, the even perfect 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]

Related numbers

• 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).

References

1. Deza, Elena (2021), Mersenne Numbers And Fermat Numbers, World Scientific, p. 263, ISBN 978-9811230332
2. 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