Consecutive integers in the form $a^x+y^b$

Let $a,b$ and $k$ be integers greater than $1$. For a tuple of $k$ consecutive integers sorted in ascending order, denoted by $T_k$, call $T_k$ a nice $k$-tuple if each integer of $T_k$ is a sum of two powers of the form $a^x+y^b$ and a perfect $k$-tuple if each integer of $T_k$ is a sum of two perfect powers of the form $a^x+y^b$, respectively. Let $N_k(a,b)$ be the number of nice $k$-tuples and ${\tilde{N}}_k(a,b)$ be the number of perfect $k$-tuples. For a given $(a,b)$, it is quite interesting to find out $N_k(a,b)$ and ${\tilde{N}}_k(a,b)$. In 2020, Lin and Cheng obtained the formula for $N_k(2,2)$. The main goal of this paper is to establish the formulas for $N_k(a,b)$ and ${\tilde{N}}_k(a,b)$. Actually, by using the method of modulo coverage together with some elementary techniques, the formulas for ${\tilde{N}}_k(2,2)$, ${\tilde{N}}_k(3,2)$ and $N_k(3,2)$ are derived.