GCD>1 based sequence

Leroy Quet qq-quet at mindspring.com
Fri Dec 5 01:53:29 CET 2003

Related to the EKG sequence, anyway, are these three interesting links:



(and this one from Neil Sloane and coauthors:)


I wonder if any of the mentioned EKG-sequence conjectures/theorems have 
analogs with my sequence.
(If I did not err, the fact that my sequence is a permutation of all 
integers >= 2 was much easier to prove than proving the same for the EKG 
sequuence, however.)

Leroy Quet 
Matthew Vandermast wrote: 
>>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 
>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...)

PS:  (recall: my sequence:

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

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