Ppeerrffeecctt Ssqquuaarreess

For a sequence of integers $A_1, A_2, \ldots, A_N$, let $G(A)$ denote a graph with $N$ nodes (numbered $1$ through $N$) such that for each pair of distinct nodes $i$ and $j$, there is an undirected edge between these nodes if and only if $A_i \cdot A_j$ is a *perfect square*. You are given a graph with $N$ nodes (numbered $1$ through $N$) and an integer $K$. Find the number of integer sequences $