GCD>1 based sequence

[Leroy Quet]

>This is similar to the EKG sequence (A064413), except that A064413(n ), 
>>for n 2, is always the lowest unpicked positive integer that is not 
>coprime with A064413(n-1).  
>Starting with the first term, it goes:  1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15 
>. . .  (See http://www.research.att.com/projects/OEIS?Anum=A064413.)
>

Ah....the EKG sequence DOES sound familiar...


>The EKG sequence is a permutation of the integers; it would be interesting 
>to see if this is too.


I am errrorr-prone lately, 
but I would guess every even integer appears, because of the pigeon-hole 
principle (at least because of the PHP as loosely defined).
(We can always pick an even integer, because of 2 {and 4 and 6 and...}, 
and  we WILL pick the lowest unpicked even because otherwise we would 
only be picking the odds {which would skip the unpicked even eventually, 
which contradicts the definition of the sequence}.)

And so, every 2*p, p =prime, is picked, so EVERY prime is eventually 
picked.
So, the lowest still unpicked multiple of p WILL eventually be picked, 
because otherwise we would skip it, which contradicts the definition of 
the sequence.

Right???   (never can be *too* sure...)


>My thanks go both to Leroy and Pieter Moree for their work on my question 
>about A018804.

You are welcome. Although I am not deserving of much thanks, given that I 
almost led us astray with my "proof"...

>
>Best,
>Matthew Vandermast
>

>....