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The EKG sequence is the sequence $a(n)$, $n\in \Bbb N_{\ge0}$ 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 $10$.

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 $10$.

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.

The EKG sequence is the sequence $a(n)$, $n\in \Bbb N_{\ge0}$ 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 $10$.

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 conscutive terms is 1, a prime or a prime power.

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

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.