Distinct subsequences

Popcount $p(x)$ of a number $x$ is defined as the number of set bits in $x$. For any sequence $s (s1,s2,s3...sk)$ of size $k$, let $f(s)$ be defined as the sum of popcount of all its elements. More formally, $$f(s)= \sum_{i=1}^kp(si)$$. A positive integer N is given . N can be split into several positive integers that can be represented as sequence $u (u1,u2,u3...um)$ of arbitrary size $m$. Form