A GCD-related sequence...update

[Leroy Quet]

I posted here regarding this sequence a couple week back.

I will just copy/paste my sci.math post (in part) on the sequence with my 
recent reply:


Leroy




> Consider this sequence...
> 
> 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)}.
> (But these definitions are obvious, I trust.)
> 
> (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}.)
> 

>....



> Any more that can be said about this sequence???
> 
> 
> thanks,
> Leroy 
>  Quet


First, my sequence is now at

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A
num=A089088

Well, if we let a(1) = n, integer >= 2 {and not necessarily 2}, we can 
get the sequence {a_n(m)}.

If we have n = odd prime,

 a_n(m)  = a_2(m-2)

for all m where 3<=m<=M,

where M is defined =from
 a_2(M-1) = 2n.


So, the limit of {a_n(m)}, as n = primes -> oo, is {n,2n,a_2(m-2)}.


But I have not discovered anything about the sequence if n is composite.