  This paper introduces a new type of covering process that covers the set of
natural numbers using renewal processes as objects. Inspired by the behavior of
prime numbers, the model in each step finds the smallest vacant point, $k$, and
place, starting in $k$, a renewal process with a step distribution given by a
geometric random variable with parameter $\frac{1}{k}$. The model depends on
its entire past, and small perturbations in its initial value can lead to very
different outcomes. Here, we expose a technique that finds the first-order
limit behavior for the number of objects placed until $n$, which exhibits
intriguing similarities to prime number distributions, having a concentration
around $n\log{n}$.
