  For a non negative measure $\mu$ with $p$ atoms, we study the relation
between the Square Root Problem of $\mu$ and the problem of subnormality of
${\tilde W_\mu}$ the Aluthge transform of the associated unilateral weighted
shift. We use an approach based on uniquely represented elements in the support
of $\mu*\mu$. We first show that if ${\tilde W_\mu}$ is subnormal, then
$2p-1\le card(supp(\mu*\mu))\le [\frac{(p-1)^2+6}{2}]$. We rewrite several
results known for finitely atomic measure having at most five atoms and give a
complete solution for measures six atoms.
