Does this sequence of product of two primes always exist

Question

Consider a sequence $A: a_1, a_2, a_3 .. a_n$ such that each element of this sequence is product of two primes from $p_1, p_2 .. p_m$ so that:

1. Each $a_i$ is distinct.

2. $m$ is the least integer so that $\frac{m(m-1)}{2} \ge n$

3. Any three consecutive elements of sequence are coprime, but any two are not.

For example,

1. for $n = 3$, we have $m = 3$ and so taking the 3 primes as $a,b,c$, we can say the sequence $A$ is $ab, bc, ac$

2. For $n = 4$, $m = 4$ and $A: ab, bc, cd, da$

3. For $n = 6$, $m = 4$ and $A: ab, bd, da, ac, cd, db$

The question is to show that such a sequence exists for any $n$.

If this sequence exists for all $n$, is there a simple pattern for $A$ ?\

Edit

There was an error in example 3, no such sequence exists for $n=6$.

Answer 1

Your example for $n=6$ doesn't work, because the terms $bd$ and $db$ are the same number, and in fact no valid sequence exists in this case.

This is really a graph theory question: terms of the sequence correspond to edges of the complete graph $K_m$ and the conditions require that two consecutive edges share a vertex but three consecutive edges don't. Equivalently, we are looking for a trail of length $n$. If $m$ is odd there will always be such a trail (because $K_m$ is Eulerian), but if $m$ is even and $n$ is close to the maximum value it won't exist.