  Let $G=(V,E)$ be a simple undirected graph. $G$ is a circulant graph defined
on $V=\mathbb{Z}_n$ with difference set $D\subseteq
\{1,2,\ldots,\lfloor\frac{n}{2}\rfloor\}$ provided two vertices $i$ and $j$ in
$\mathbb{Z}_n$ are adjacent if and only if $\min\{|i-j|, n-|i-j|\}\in D$. For
convenience, we use $G(n;D)$ to denote such a circulant graph.
  A function $f:V(G)\rightarrow\mathbb{N}\cup\{0\}$ is an integer
$\{k\}$-domination function if for each $v\in V(G)$, $\sum_{u\in
N_G[v]}f(u)\geq k.$ By considering all $\{k\}$-domination functions $f$, the
minimum value of $\sum_{v\in V(G)}f(v)$ is the $\{k\}$-domination number of
$G$, denoted by $\gamma_k(G)$. In this paper, we prove that if
$D=\{1,2,\ldots,t\}$, $1\leq t\leq \frac{n-1}{2}$, then the integer
$\{k\}$-domination number of $G(n;D)$ is $\lceil\frac{kn}{2t+1}\rceil$.
