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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.

1

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.