  Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written
in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots,
n_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the
minimum of $(n_1+\cdots+n_l)/\ord(g)$ over all possible $g\in G$ such that
$\langle g \rangle =G$. An open problem on the index of length four sequences
asks whether or not every minimal zero-sum sequence of length 4 over a finite
cyclic group $G$ with $\gcd(|G|, 6)=1$ has index 1. In this paper, we show that
if $G=\langle g\rangle$ is a cyclic group with order of a product of two prime
powers and $\gcd(|G|, 6)=1$, then every minimal zero-sum sequence $S$ of the
form $S=(g)(n_2g)(n_3g)(n_4g)$ has index 1. In particular, our result confirms
that the above problem has an affirmative answer when the order of $G$ is a
product of two different prime numbers or a prime power, extending a recent
result by the first author, Plyley, Yuan and Zeng.
