[seqfan] Re: No 3 terms in Geom Progression
israel at math.ubc.ca
israel at math.ubc.ca
Thu Mar 1 21:20:10 CET 2012
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
>
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