[seqfan] Re: No 3 terms in Geom Progression

[israel at math.ubc.ca]

I confirm David's numbers and extend the sequence to n=100:

1, 2, 3, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 13, 14, 14, 15, 15, 16, 
17, 18, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 29, 30, 31, 
32, 33, 34, 34, 35, 36, 37, 38, 38, 38, 39, 39, 40, 41, 42, 43, 44, 45, 46, 
46, 47, 48, 49, 49, 50, 51, 52, 52, 53, 54, 55, 55, 56, 57, 57, 57, 58, 59, 
60, 61, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 71, 72, 73, 74, 74, 75, 
75, 75, 75

Here is a Maple program for computing the n'th term.

A:= proc(n)
       local cons, x;
       cons:=map(op,{seq(map(t -> x[t]+x[b]+x[b^2/t]<=2,
            select(t -> (t<b) and (t>=b^2/n),
         numtheory:-divisors(b^2))),b=2..n-1)});
       Optimization:-Maximize(add(x[i],i=1..n),cons, assume=binary)[1]
 end proc;

Robert Israel
University of British Columbia

On Mar 1 2012, David Applegate wrote:

>1..16 \ {1,4} still contains 3,6,12 and 9,12,16.
>
>I believe the corrected and extended sequence is:
>
>  
> 1,2,3,3,4,5,6,7,7,8,9,10,11,12,13,13,14,14,15,15,16,17,18,19,19,20,21,21,22,23, 
> 24,24,25,26,27,27,28,29,30,31,32,33,34,34,35,36,37,38,38,38,39,39,40,41,42,43, 
> 44,45,46,46,47,48,49,49,50,51,52,52,53,54,55,55,56,57,57,57,58,59,60,61,61,62
>
>However, my computation used a floating-point IP solver for the
>packing subproblems, so although it's almost certainly correct I
>wouldn't bet my life on it.
>
>My approach was to enumerate geometric progressions using
>
>  for (i=1;i<=N;i++) {
>    for (j=2; j*j<=i; j++) {
>      if (i % (j*j) != 0) continue;
>      for (k=1; k<j; k++) {
>	print i*k*k/(j*j), i*k/j, i;
>      }
>    }
>  }
>
>and then solve the integer program of maximizing the subset of {1..N}
>subject to not taking all 3 of any progression.
>
>-Dave
>
>> From seqfan-bounces at list.seqfan.eu Thu Mar  1 12:01:22 2012
>> Date: Thu, 1 Mar 2012 12:01:05 -0500
>> From: Neil Sloane <njasloane at gmail.com>
>> To: seqfans <seqfan at seqfan.eu>
>> Subject: [seqfan] No 3 terms in Geom Progression
>
>> *Dear Sequence Fans,* *A003002 gives the size of the largest subset of 
>> [1,2,...,n] which contains no* *3-term arithmetic progression.* *But 
>> what is ** he largest subset of [1,2,...,n] which contains no * *3-term 
>> geometric progression?* *E.g. if n=16, it looks like omitting 1 and 4 
>> works, so a(16) = 14* *A quick hand calculation gives (for n>=1):* 
>> *1,2,3,3,4,5,6,7,7,8,9,10,11,12,13,14* * * *Could someone correct/extend 
>> this?* *Neil*
>
>> -- 
>> Dear Friends, I will soon be retiring from AT&T. New coordinates:
>
>> Neil J. A. Sloane, President, OEIS Foundation
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA
>> Phone: 732 828 6098; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>
>> _______________________________________________
>
>> Seqfan Mailing list - http://list.seqfan.eu/
>
>
>_______________________________________________
>
>Seqfan Mailing list - http://list.seqfan.eu/
>