It is known that Almost all primes have a multiple of small Hamming weight, which makes me wonder what is known about the least multiples of primes that have least Hamming weight.

Questions:

• what are the values of the sequence $\mu(n)\ :=\ \min\limits_{m\in\mathbb{N}}: p_nm=\min\limits_{d_i\in\lbrace0,1\rbrace}\sum\limits_{i=0}^\infty d_i2^i$,
  where $p_n$ is the $n$-th prime number?
  $\mu(1)=1,\ \mu(2)=1,\ \mu(3)=1,\ \mu(4)=1,\ \mu(5)=3,\ \mu(6)=5,\ \dots$

• for which $n$ is $H(p_n\mu(n)) \gt 6$ if $H()$ is denotes the Hammig weight of the argument ?