Generalized LCM matrices

Let $f$ be an arithmetical function. The matrix $[f[i,j]{]}_{n\times n}$ given by the value of $f$ in least common multiple of $[i,j]$, $f([i,j])$ as its $i,j$ entry is called the least common multiple (LCM) matrix. We consider the generalization of this matrix where the elements are in the form $f(n,[i,j])$ and $f(n,i,j,[i,j])$.