The EKG sequence is the sequence where a(1)=1, a(2)=2, a(n)=least positive integer not in {a(1),a(2),...,a(n-1)} that has a non-trivial common factor with a(n-1).
Conjecture: The gcd between any two consecutive terms is 1, a prime or a prime power.
Counterexample: a(578)=620, a(579)=610; the gcd is 10.