  In this paper we study almost $p$-ary sequences and their autocorrelation
coefficients. We first study the number $\ell$ of distinct out-of-phase
autocorrelation coefficients for an almost $p$-ary sequence of period $n+s$
with $s$ consecutive zero-symbols. We prove an upper bound and a lower bound on
$\ell$. It is shown that $\ell$ can not be less than $\min\{s,p,n\}$. In
particular, it is shown that a nearly perfect sequence with at least two
consecutive zero symbols does not exist. Next we define a new difference set,
partial direct product difference set (PDPDS), and we prove the connection
between an almost $p$-ary nearly perfect sequence of type $(\gamma_1,
\gamma_2)$ and period $n+2$ with two consecutive zero-symbols and a cyclic
$(n+2,p,n,\frac{n-\gamma_2 - 2}{p}+\gamma_2,0,\frac{n-\gamma_1
-1}{p}+\gamma_1,\frac{n-\gamma_2 - 2}{p},\frac{n-\gamma_1 -1}{p})$ PDPDS for
arbitrary integers $\gamma_1$ and $\gamma_2$. Then we prove a necessary
condition on $\gamma_2$ for the existence of such sequences. In particular, we
show that they don't exist for $\gamma_2 \leq -3$.
