This is not an answer but an observation.
Let $n$ be an integer and $H(n)$ its Hamming weight.
$H(n) <= 1+ \max\( \{ H(d) |\ d\ {\rm divisor\ of\ } n-1 \}\)$
in particular for $p$ a prime greater than 2
$H(p) <= 1+ \max( \{ H(d) |\ d\ {\rm proper\ divisor\ of }\ p-1 \})$
It could suggest ways to attack this and related questions.