So we are trying to find two sets of positive integers $\lbrace a_{1},...,a_{n} \rbrace$ and $\lbrace b_{1},...,b_{n}\rbrace$, with nomember of one set in the other, such that $$\sum_{k=1}^{n} a_{k} = \sum_{k=1}^{n} b_{k}$$$$\sum_{k=1}^{n} a_{k}^{2} = \sum_{k=1}^{n} b_{k}^{2}$$ $$\vdots$$ $$\sum_{k=1}^{n} a_{k}^{N} = \sum_{k=1}^{n} b_{k}^{N}$$ For some integer $r$. I define 'smallest' in the question title as the solution set with smallest value of $\sum_{k=1}^{n} a_{k}$.
For example, for $N=1$ the solution is, $\lbrace 3,1\rbrace$ and $\lbrace 2,2 \rbrace$ as there is only one way to sum $2$ and $3$. For $N=2$, I believe the smallest solution is $\lbrace 4,4,1\rbrace$ and $\lbrace 5,2,2 \rbrace$. How can I find solutions for higher values of $N$?