The EKG sequence 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 $10$.