  We present an optimization procedure for a seminal class of positive maps
$\tau_{n,k}$ in the algebra of $n \times n$ complex matrices introduced and
studied by Tanahasi and Tomiyama, Ando, Nakamura and Osaka. Recently, these
maps were proved to be optimal whenever the greatest common divisor
$GCD(n,k)=1$. We attain a general conjecture how to optimize a map $\tau_{n,k}$
when $GCD(n,k)=2$ or 3. For $GCD(n,k)=2$, a series of analytical results are
derived and for $GCD(n,k)=3$, we provide a suitable numerical analysis.
