On zero-sum problems of two new types

Abstract: In this paper, we mainly investigate zero-sum problems over $\mathbb Z/n\mathbb Z$ (with $n&gt;1$) of two new types. Let $s_1(n)$ (resp. $t_1(n)$) be the least positive integer $k$ such that for any integers $a_1,\ldots,a_k$ not divisible by $n$ (resp., relatively prime to $n$), there is an $I\subseteq{1,\ldots,k}$ with $|I|=n$ for which the sum $\sum_{i\in I}a_i$ is divisible by $n$ but not divisible by $n<sup>2$. For $n\geqslant 4$, we prove that $2n+1\leqslant s_1(n)\leqslant n<sup>2-2n+2$ and $$2n-(-1)^n\leqslant t_1(n)\leqslant (n-1)\varphi(n)+1$$. We conjecture that $s_1(n)=2n+1$ and $t_1(n)=2n-(-1)<sup>n$ for any integer $n&gt;2$.