pernicious number
{{Short description|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.{{citation|first=Elena|last=Deza|author-link=Elena Deza|title=Mersenne Numbers And Fermat Numbers|page=263|publisher=World Scientific|year=2021|isbn=978-9811230332}}
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).{{cite OEIS|A052294|mode=cs2}} The sequence of pernicious numbers begins
{{bi|left=1.6|3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 24, ... {{OEIS|A052294}}.}}
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 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.
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.{{citation|first1=Simon|last1=Colton|first2=Louise|last2=Dennis|title=Seventh International Symposium on Artificial Intelligence and Mathematics|contribution=The NumbersWithNames Program|year=2002|contribution-url=https://nottingham-repository.worktribe.com/output/1022768}}{{citation|first=Tianxin|last=Cai|author-link=Tianxin Cai|title=Perfect Numbers And Fibonacci Sequences|page=50|publisher=World Scientific|year=2022|isbn=978-9811244094}}
Related numbers
- Odious numbers are numbers with an odd number of 1s in their binary expansion ({{OEIS2C|id=A000069}}).
- Evil numbers are numbers with an even number of 1s in their binary expansion ({{OEIS2C|id=A001969}}).
References
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{{Classes of natural numbers |state=collapsed}}