Example
N = 9 and K=3
4 + 4 + 1 = 9 .
What I have tried.
We can not go on dividing with 2.
We can use unbounded knapsack with array elements from 2^0 to 2^32.
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If $k>N$, this cannot be done (as $\forall p_1,\dots, p_k\geq 0,\sum_{i=1}^k 2^{p_i}\geq k>N$)
If $k\leq N$: Write $N$ in binary, and denote by $n$ the number of $1$s. We have written $N$ as a sum of $n$ powers of two (which happen to all be distinct).
- If $n\leq k$, then you can always split some of the powers of two (iteratively if needed) in order to obtain $N$ as a sum of $k$ powers of two (you cannot be stuck with only terms of the form $2^0$ -which cannot be split- before reaching $k$ terms as $k\leq N$ by assumption).
- If $n>k$, then it cannot be done. Indeed, assume it could be done, and consider the decomposition into $k$ powers of two. (Iteratively) merge these powers of two until you reach a binary decomposition of $N$. This decomposition has at most $k$ $1$s, which is absurd by uniqueness of the binary decomposition.
