FInd a set of $n$ numbers to minimize number of distinct products I would liked to find a set of $n$ natural numbers none of which are perfect squares that minimizes the number of distinct products of any two numbers in the set. 

Stated more formally: find a generating set $S$ of $n$ non-perfect square natural numbers such that the number of unique products of any pair of numbers in the set $S$ is minimized. We only care about products of distinct elements in $S$.
According to Cauchy–Davenport, you always have at least $2n-1$ distinct products, and this is achieved by geometric series: $a,ab,\ldots,ab^{n-1}$. Now see if you can arrange that none of these numbers is a perfect square.
According to Cauchy–Davenport, you always have at least $2n-1$ distinct products, and this is achieved by geometric series: $a,ab,\ldots,ab^{n-1}$. Now see if you can arrange that none of these numbers is a perfect square.