In this answer which I wrote a few days ago, I posit that for any polynomial $P \in \mathbb{N}[x]$, the asymptotic density of the set of natural numbers $n$ such that $P(n)$ is odious (that is; has an odd number of 1 bits in its binary expansion) is $\frac{1}{2}$, and likewise for evil numbers (t...

An evil number is a positive integer $n$ that has an even number of $1$s in its binary expansion. Many theorems exist about evil numbers, the most known ones are probably those that involve the Thue-Morse sequence. However, I find no information about prime numbers having an even number of $1$s...