  Assume $x,\ y,\ n$ are positive integers and $n$ is odd. In this note, we
show that the class number of the imaginary quadratic field
$\mathbb{Q}(\sqrt{x^{2}-y^{n}})$ is divisible by $n$ for fixed $x, n$ if
$\gcd(2x,y)=1$ and $y>C$ where $C$ is a constant depending only on $x$ and $n$.
Based on this result, for any odd integer $n$ and any positive integer $m$, we
construct an infinite family of $m+1$ successive imaginary quadratic fields
$\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1^{2}})$, $\cdots$,
$\mathbb{Q}(\sqrt{d+m^{2}})$ $(d\in \mathbb{Z})$ whose class numbers are all
divisible by $n$.
