pernicious number

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}}.}}

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 $n>1$, 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.

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.{{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}}

• 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}}).

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{{Classes of natural numbers |state=collapsed}}

Category:Base-dependent integer sequences