For example,    is odious since the sum of its binary digits is 3.

For  2$">, the number of evil and odious numbers of with    binary digits is the same, and the two sets have also the same sum, i.e.,  .

A few numbers which are equal to the sum of their odious proper divisors are known. They are 28, 496, 8128, 415800, 2096128, 33550336, and 8589869056. They could be called odious-perfect numbers and it is worth to note that two of them, 415800 and 2096128, are not perfect numbers.

The first odious numbers are 1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, 41, 42, 44, 47, 49, 50, 52, 55 more terms

Below, the spiral pattern of odious numbers up to 2500. See the page on prime numbers for an explanation and links to similar pictures.

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many odious numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.