Let $\mathbb N_{n} = \{1,2,\cdots,n\}$.

Let $S$ be of cardinality $n$ where elements of $S$ are integers from $\mathbb N_{n}$ and at least one element of $S$ is repeated (That is at least one integer from $\mathbb N_{n}$ is skipped. One can easily find a set $S$ with the property that: $\displaystyle \sum_{j \in S}j^{i} = \displaystyle \sum_{j \in \mathbb N_{n}}j^{i}$ when $i = 1$. (Example: $n=4$, $S=\{1,1,4,4\}$ has sum $10$, the same as sum of first n consecutive integers)

How about for $i \ge 2$? It is not obvious that higher power sum sets exist due to constraint in the cardinality of $S$ and $\mathbb N_{n}$. One cannot deny it either? Is there a easy way to tackle some sumset questions?

For $i=2$ it is related to quadratic forms and integer norms. In an integer coordinate system, how many ways can a given integer norm occur when the coordinates are bounded?