evil	odious
The first 16 evil and odious numbers in binary. It can be seen, that both sequences differ only in the least significant bits, which form the Thue–Morse sequence for the evil, and its negation for the odious numbers. The other bits form the even numbers.

Innumber theory, anevil number is a non-negative integer that has an evennumber of 1s in itsbinary expansion.^[1] These numbers give the positions of the zero values in theThue–Morse sequence, and for this reason they have also been called theThue–Morse set.^[2] Non-negative integers that are not evil are calledodious numbers.

The first evil numbers are:

0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24, 27, 29, 30, 33, 34, 36, 39 ...^[1]

The partition of the non-negative integers into the odious and evil numbers is the unique partition of these numbers into two sets that have equalmultisets of pairwise sums.^[3]

As 19th-century mathematician Eugène Prouhet showed, the partition into evil and odious numbers of the numbers from${\displaystyle 0}$ to${\displaystyle 2^k-1}$, for any${\displaystyle k}$, provides a solution to theProuhet–Tarry–Escott problem of finding sets of numbers whose sums of powers are equal up to the${\displaystyle k}$th power.^[4]

Incomputer science, an evil number is said to haveeven parity.

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