An improvement of Prouhet's 1851 result on multigrade chains

In 1851 Prouhet showed that when $N=j^{k+1}$ where $j$ and $k$ are positive integers, $j \geq 2$, the first $N$ consecutive positive integers can be separated into $j$ sets, each set containing $j^k$ integers, such that the sum of the $r$-th powers of the members of each set is the same for $r=1,\,2,\,\ldots,\,k$. In this paper we show that even when $N$ has the much smaller value $2j^k$, the first $N$ consecutive positive integers can be separated into $j$ sets, each set containing $2j^{k-1}$ integers, such that the integers of each set have equal sums of $r$-th powers for $r=1,\,2,\,\ldots,\,k$. Moreover, we show that this can be done in at least $\{(j-1)!\}^{k-1}$ ways. We also show that there are infinitely many other positive integers $N=js$ such that the first $N$ consecutive positive integers can similarly be separated into $j$ sets of integers, each set containing $s$ integers, with equal sums of $r$-th powers for $r=1,\,2,\,\ldots,\,k$, with the value of $k$ depending on the integer $N$.