I have come over the following problem in elementary number theory: suppose that a positive integer $k$ is given. Is it possible to find, for every such $k$, a nonnegative integer $N(k)$ and a set of nonnegative integers $S(k) = \{s_1,\ldots,s_k\}$ of cardinality $k$, such that $N(k) = s_1 + \ldots + s_k$ and such that this is a unique partition of $N(k)$ with parts taken from $S(k)$ (that is there is not any other partition of $N(k)$ with parts taken from $S(k)$)? For instance, for $k = 3$, one may choose $N(3)= 3004$ and $S(3) = \{1000,1001,1003\}$. However, one may not choose $N(3) = 3003$ and $S(3) = \{1000,1001,1002\}$, since $1001 + 1001 + 1001 = 3003$. Or, one cannot choose $N(2) = 3$ and $S(2) = \{1,2\}$, since $3 = 1 + 1 + 1$.
I suspect that this should be possible to do for general $k$, however I have failed to characterize $N(k)$ and $S(k)$ in general.
My apologies if this problem is trivial.