Odious number

In number theory, an odious number is a positive integer that has an odd number of 1s in its binary expansion.

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 ... [1]

These numbers give the positions of the nonzero values in the Thue–Morse sequence.

Non-negative integers that are not odious are called evil numbers. 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 equal multisets of pairwise sums.[2]

If ${\displaystyle a(n)}$ denotes the ${\displaystyle n}$th odious number (with ${\displaystyle a(0)=1}$), then for all ${\displaystyle n}$, ${\displaystyle a(a(n))=2a(n)}$.[3]

In computer science, an odious number is said to have odd parity.