 Decisive number

A positive integer $n$ is decisive when every integer $a$ satisfying $1<a<n$ and $\gcd (a,n)=1$ is a prime number. There are only finitely many decisive numbers: every prime $p<\sqrt{n}$ must divide a decisive $n$, so the Chebyshev theta function gives $\log n\ge \vartheta (\sqrt{n})$, which eventually contradicts the linear lower bound for $\vartheta (\sqrt{n})$.