  Fix a set $D$ of positive integers. We study the maximum density $\mu(D)$ of
sequences of integers in which the separation between any two terms does not
fall in $D$. The $D$-sets considered in this article are of the form
$\{1,j,k\}$. The closely related function $\kappa(D)$, the parameter involved
in the "lonely runner conjecture," is also investigated. Exact values of
$\kappa(D)$ and $\mu(D)$ are found for some families of $D=\{1,j,k\}$. We prove
that the boundary conditions in two earlier results of Haralambis are sharp.
Consequently, our results declaim two conjectures posted recently, and extend
some results by Gupta.
