This is not an answer but an observation.
Let $n$ be an integer and $H(n)$ its Hamming weight.
$H(n)\le1+\max(\{ H(d) |\ d\ {\rm divisor\ of\ } n-1 \})$
in particular for $p$ a prime greater than 2
$H(p)\le1+\max( \{ H(d) |\ d\ {\rm proper\ divisor\ of }\ p-1 \})$
It could suggest ways to attack this and related questions.
