Metrics on integers I am looking for a list of distances that are defined on the set of the positive integers.
I am mostly interested in metrics that make the set complete, but I also consider other metrics.
Any reference would be appreciated!
 A: Here are some examples:
$$
\begin{align}
d(x,y)=&|x-y| \\[3mm]
d(x,y)=&|x^3-y^3|\\[3mm]
d(x,y)=& \frac{|x-y|}{1+|x-y|} \\[3mm]
d(x,y)=&|\text{arctan}(x)-\text{arctan}(y)| \\[3mm]
d_p(x,y)=&\begin{cases} p^{-v_p(x-y)} & \text{if}\quad  x\ne y \\ 0 & \text{if} \quad x=y \end
{cases},
\end{align}
$$
where $v_p(k):=\max \{\ell\in \mathbb{N}_0\; : p^\ell\; | \;k \;\}$ (This is the $p$-adic valuation). Also:
\begin{align}
d'(x,y)=f(d(x,y)),
\end{align}
where $d$ is any of the above metrics and $f:[0,+\infty) \to [0,+\infty)$ is any (not necessarily strictly) increasing and concave function such that $f(x)=0 \Leftrightarrow x=0$.
For more examples, you might want to take a look at:
Metric Preserving Functions, Dobo$\check{\text{s}}.$ J.
80-88896-30-4, Vydavatel'stvo Troffek, Koice, Slovakia (October 1998)
Google scholar
A: Any one-to-one correspondence from $\mathbb N$ to a countable complete metric space induces such a metric.  There are lots of these.
One set of examples: take any function $d$ on $\mathbb N \times \mathbb N$ that is symmetric ($d(i,j) = d(j,i)$) and satisfies $1 \le d(i,j) \le 2$ 
for all $i,j$.
