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.

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.

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.

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.