To count clean triangles we count on $imph(n)$

Abstract

A clean lattice triangle in ${\mathbb R}^2$ is a triangle that does not contain any lattice points on its sides other than its vertices. The central goal of this paper is to count the number of clean triangles of a given area up to unimodular equivalence. In doing so we use a variant of the Euler phi function which we call $imph(n)$ (imitation phi).