  Let $k\geq 3$ and $n\geq 3$ be odd integers, and let $m\geq 0$ be any
integer. For a prime number $\ell$, we prove that the class number of the
imaginary quadratic field $\mathbb{Q}(\sqrt{\ell^{2m}-2k^n})$ is either
divisible by $n$ or by a specific divisor of $n$. Applying this result, we
construct an infinite family of certain tuples of imaginary quadratic fields of
the form $$\left(\mathbb{Q}(\sqrt{d}), \mathbb{Q}(\sqrt{d+1}),
\mathbb{Q}(\sqrt{4d+1}), \mathbb{Q}(\sqrt{2d+4}), \mathbb{Q}(\sqrt{2d+16}),
\cdots, \mathbb{Q}(\sqrt{2d+4^t}) \right)$$ with $d\in \mathbb{Z}$ and $1\leq
4^t\leq 2|d|$ whose class numbers are all divisible by $n$. Our proofs use some
deep results about primitive divisors of Lehmer sequences.
