Determining which members of a set are an integeral multiple of a member of a different set

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I'm given two sets with a regular formula for the terms of each set.

$A=\left\{3,7,15,31,\cdots,2^{(n+1)}-1,\cdots,2047\right\}$

$B=\left\{3,5,9,17,\cdots,2^n+1,\cdots,2^{22}+1\right\}$

The question asks to find the sum of all distinct members of $B$ that are integral multiples of at least one member of $A$.

I noticed that every other member of $B$ is divisible by 3, although I would be interested in a mathematical way to explain why $2^{(2n+1)}+1$ is always divisible by 3. However, there's still the question of if any other members of $B$ are divisible by the other members of $A$.