The [EKG sequence](https://oeis.org/A064413) is the sequence $a(n)$, $n\in \Bbb N_{\ge1}$ where 
$$
a(n)= 
\begin{cases}
1, &n=1 \\ 
2, &n=2 \\
\begin{array}{}
\text{least positive integer not in}\\
\text{$\{a(1),a(2),\ldots,a(n-1)\}$ that } \\ 
\text{has a non-trivial common }\\
\text{factor with $a(n-1)$}
\end{array} &n>2
\end{cases}
$$

**Conjecture**: The $\gcd$ between any two consecutive terms is $1$, a prime or a prime power.

**Counterexample**: $a(578)=620$, $a(579)=610$, $\gcd$ is $10$.