BD-BDMO 2021 Problem 7

A binary string is a word containing only $0$ s and $1$ s. In a binary string, a $1-$ run is a non extendable substring containing only $1$ s. Given a positive integer $n$ , let $B(n)$ be the number of $1-$ runs in the binary representation of $n$ . For example, $B(107)=3$ since $107$ in binary is $1101011$ which has exactly three $1-$ runs. What is the following expression equal to? $$B(1)+B(2)+B(3)+ \dots + B(255)$$