  Let PR$[n]$ be the graph whose vertices are $2,3,\ldots,n$ with vertex $v$
adjacent to vertex $w$ if and only if $\gcd(v,w)>1$. It is shown that $\pi(n)$,
the the number of primes no more than $n$, equals the Lov\'{a}sz number of this
graph. This result suggests new avenues for graph-theoretic investigations of
number-theoretic problems.
