  Let $\rho=(\frac{p}{q})^{\frac{1}{r}}<1$ for some $p,q,r\in\mathbb{N}$ with
$(p,q)=1$ and $\mathcal{D}_{n}=\{0,1,\cdot\cdot\cdot,N_{n}-1\}$, where $N_{n}$
is prime for all $n\in\mathbb{N}$, and denote
$M=\sup\{N_{n}:n=1,2,3,\ldots\}<\infty$. The associated Borel probability
measure
$$\mu_{\rho,\{\mathcal{D}_{n}\}}=\delta_{\rho\mathcal{D}_{1}}*\delta_{\rho^{2}\mathcal{D}_{2}}*\delta_{\rho^{3}\mathcal{D}_{3}}*\cdots$$
is called a Moran measure. Recently, Deng and Li proved that
$\mu_{\rho,\{\mathcal{D}_{n}\}}$ is a spectral measure if and only if
$\frac{1}{N_{n}\rho}$ is an integer for all $n\geq 2$. In this paper, we prove
that if $L^{2}(\mu_{\rho, \{\mathcal{D}_{n}\}})$ contains an infinite
orthogonal exponential set, then there exist infinite positive integers $n_{l}$
such that $(q,N_{n_{l}})>1$. Contrastly, if $(q,N_{n})=1$ and $(p,N_{n})=1$ for
all $n\in\mathbb{N}$, then there are at most $M$ mutually orthogonal
exponential functions in $L^{2}(\mu_{\rho, \{\mathcal{D}_{n}\}})$ and $M$ is
the best possible. If $(q,N_{n})=1$ and $(p,N_{n})>1$ for all $n\in\mathbb{N}$,
then there are any number of orthogonal exponential functions in
$L^{2}(\mu_{\rho, \{\mathcal{D}_{n}\}})$.
