Can we arrange the first n natural numbers consecutive numbers can't be split in equal sums

Question

For some set S, define a reflection as a subset of consecutive elements that can be split in a way so that the sums of both sides are equal. The set ${1,2,...,7,8} $ has reflections $1+2 =3$ and $4+5+6=7+8$. Can we rearrange $S_n=1,.2,..,n$ in a way so there are no reflections for arbitrary large n? For example, the arrangement ${8,2, 1,4,6,5,3,7}$ would be such an arrangement of $S_8$.

Answer 1

When neither $n$ nor $n-1$ belongs to $T$ = A001652, then we can build a permutation $S_n$ of $1..n$ with the desired property as follows:

• if $k \in T$ then $S_n(k) = k+1$,
• if $k-1 \in T$ then $S_n(k) = k-1$,
• otherwise $S_n(k) = k$.

By design, no prefix of $S_n$ can be split into two parts with equal sum, and $n$ can be arbitrary large.

To the limit, we can build a permutation $S$ of the positive integers where no prefix has reflection.