GCD>1 based sequence

[Leroy Quet]

Unexpectedly, this sequence is not now in the EIS:

2, 4, 6, 3, 8, 9, 10, 5, 12, 14, 7, 15, 16, 18, 20,...


a(1) = 2;
a(m) = lowest unpicked positive integer which is *not* coprime with at 
least one previous term of the sequence.

By "unpicked", I mean the integer is not among {a(1),...,a(m-1)}.
And by "not coprime with at least one previous term", I mean that at 
least one prime dividing a(m) also divides at least one element of 
{a(1),...,a(m-1)}.
(By these definitions are obvious, I trust.)


I could have sworn this sequence was at one time in the EIS. Am I 
thinking of something similarly defined, or was the sequence eliminated 
because each term was, say, 1/2 of another sequence's term? (I have not 
put this into Superseeker.)

 
It seems like this (new?) sequence might be a permutation of the integers 
>= 2.

(It might be advantageous to define a(0) = 1 for whatever reason. {I am 
sure there is a good reason for doing such, but I am unsure now why}.)

thanks,
Leroy Quet