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==
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. ==Related numbers==
Related numbers
• Odious numbers are numbers with an odd number of 1s in their binary expansion (). • Evil numbers are numbers with an even number of 1s in their binary expansion (). ==References==