Let $a,n$ be positive integers and $S=\{x_1,...,x_n\}$ be a set of $n$ distinct positive integers. The set $S$ is said to be gcd (resp. lcm) closed if $gcd(x_i,x_j)\in S$ (resp. $[x_i,x_j]\in S$) for all integers $i,j$ with $1\le i,j\le n$. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix having the $a$th power of the greatest common divisor (resp. the least common multiple) of $x_i$ and $x_j$ as its $(i,j)$-entry. In this paper, we mainly show that for any positive integer $a$ with $a\ge 2$, the power LCM matrix $[S^a]$ defined on a certain class of gcd-closed (resp. lcm-closed) sets $S$ is nonsingular. This provides evidences to a conjecture raised by Shaofang Hong in 2002.

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.