This is not an answer but an observation.

Let $n$ be an integer and $H(n)$ its Hamming weight.

$H(n) <= 1+ \max\( \{ H(d) | d divisor of n-1 \}\)$

in particular for $p$ a prime greater than 2

$H(p) <= 1+ \max\( \{ H(d) | d proper divisor of p-1 \}\)$

It could suggest ways to attack this and related questions.