Let $s_n$ be the largest size of a set $S$ of integers $\ge0$ st there exist two subsets $A,B\subseteq \{1,2,...,n\}$ that satisfy the conditions
(1) $A\cap B=\emptyset$,
(2) $A\cup B=\{1,2,...,n\}$,
(3) $\sum_\limits{a\in A}a^k=\sum_\limits{b\in B}b^k$ for all $k\in S$.
What is $s_n$? What is an example for $S,A,B$ in the case $|S|=s_n$?
This question is inspired by a question my teacher asked me. The original question was to show that $\{1,2,...,2^m\}$ can be partitioned into two sets $A,B$ of equal size st $\sum_\limits{a\in A}a^k=\sum_\limits{b\in B}b^k$ for $k=0,1,...,(m-1).\quad$ I solved that problem (by induction on m) and wonder about what happens if I replace $2^m$ by other integers $n$.
First $s_n=0$ for all $n\equiv 1\pmod{4}$ because the number of odd numbers in $\{1,2,...,n\}$ is odd. Then $s_n=1$ for all $n\equiv 2\pmod{4}$ using the same reason and only $S=\{0\}$ works. For $n=2^mt\space$ where $t$ is odd and $m\ge 2$, I know from the question my teacher asked me that $s_n\ge m$ because $S=\{0,1,...,m-1\}$ is an example. I'm not sure whether $s_n=m$ in these cases. Please help me solve the case $n=2^mt\space$ where $t$ is odd and $m\ge 2$.
