Consider the set $A$ of natural numbers which are of the form $k^3-p$, for $k$ a positive integer and $p$ a positive prime. Does $A$ have a density (of any of the usual kinds for sets of natural numbers) and if so, what is it?
$A$ contains a lot of numbers, and it seems somewhat difficult to me to prove that a particular number is not in $A$, except that most cubes are not in $A$.