  The Piatetski-Shapiro sequences are of the form $\mathcal{N}^{(c)} :=
(\lfloor n^c \rfloor)_{n=1}^\infty$ with $c > 1, c \not\in \mathbb{N}$. In this
paper, we study the distribution of pairs $(p, p^{\#})$ of consecutive primes
such that $p \in \mathcal{N}^{(c_1)}$ and $p^{\#} \in \mathcal{N}^{(c_2)}$ for
$c_1, c_2 > 1$ and give a conjecture with the prime counting functions of the
pairs $(p, p^{\#})$. We give a heuristic argument to support this prediction
which relies on a strong form of the Hardy-Littlewood conjecture. Moreover, we
prove a proposition related to the average of singular series with a weight of
a complex exponential function.
