  We define a new metric between natural numbers induced by the $\ell_\infty$
norm of their unique prime signatures. In this space, we look at the natural
analog of the number line and study the arithmetic function $L_\infty(N)$,
which tabulates the cumulative sum of distances between consecutive natural
numbers up to $N$ in this new metric.
  Our main result is to identify the positive and finite limit of the sequence
$L_\infty(N)/N$ as the expectation of a certain random variable. The main
technical contribution is to show with elementary probability that for $K=1,2$
or $3$ and $\omega_0,\ldots,\omega_K\geq 2$ the following asymptotic density
holds $$ \lim_{n\to\infty}\frac{\big|\big\{M\leq n:\; \|M-j\|_\infty <\omega_j
\text{ for } j=0,\ldots,K \big\}\big|}{n} = \prod_{p:\, \mathrm{prime}}\!
\bigg( 1- \sum_{j=0}^K\frac{1}{p^{\omega_j}} \bigg)~. $$ This is a
generalization of the formula for $k$-free numbers, i.e. when
$\omega_0=\ldots=\omega_K=k$. The random variable is derived from the joint
distribution when $K=1$.
  As an application, we obtain a modified version of the prime number theorem.
Our computations up to $N=10^{12}$ have also revealed that prime gaps show a
considerably richer structure than on the traditional number line. Moreover, we
raise additional open problems, which could be of independent interest.
