[seqfan] Re: Phil Scovis's problem

israel at math.ubc.ca israel at math.ubc.ca
Wed Mar 6 01:03:08 CET 2013


Also b(8) = 480 with S = {135,252,270,320,336,360,448,480}.  The fact
that {320,336,360,448,480} = 2*{160,168,180,224,240} = 4*{80,84,90,112,120}
suggests a conjecture that b(n) = 15 * 2^(n-3) for n >= 6.
I can also report that b(9) <= 960 with a possible S being
{135,378,504,540,640,672,720,896,960}, though I haven't 
ruled out b(9) < 960.

Robert Israel
University of British Columbia 


On Mar 5 2013, israel at math.ubc.ca wrote:

>I get the following for b(n):
>
>n   b(n)    one S
>------------------
>1    1       {1}
>2    2       {1,2}
>3    6       {2,3,6}
>4   18       {4,9,12,18}
>5   54       {8,24,27,36,54}
>6  120       {45,80,84,90,112,120}
>7  240       {45,126,160,168,180,224,240} 
>
>The sequence doesn't appear to be in the OEIS.
>
>Robert Israel
>University of British Columbia
>
>On Mar 5 2013, Neil Sloane wrote:
>
>>Dear Sequence Fans:
>>
>>If you look at the History tab for A213909 you see the following problem,
>>studied by Phil Scovis:
>>
>>Definition. Let S be a set of n positive numbers such that
>>all n choose 2 pairwise GCD's are distinct, and let
>>m(S) (resp. M(S)) denote the smallest and greatest elements of S;
>>a(n) is the minimal value of m(S) over all choices for S.
>>
>>and a second sequence,
>>
>>b(n) is the minimal value of M(S) over all choices for S.
>>
>>Example: For n=4, S = {4,9,12,18} has its six GCD's equal to
>>1,4,2,3,9,6, so it satisfies the condition, and shows that
>>a(4) <= 4, b(4) <= 18.
>>But S = {8,9,10,12} is not legal, since GCD(8,9) = 1 = GCD(9,10), and the
>>GCD's are not all distinct.
>>
>> The values that were submitted - probably intended to be the b(n) 
>> sequence - don't look right, and the submitter, perhaps wisely, withdrew 
>> the sequence.
>>
>>But the questions seem interesting. What are the a(n) and b(n) sequences,
>>and are they in the OEIS?
>>(The closest entry I can find is Alois Heinz's A196719.)
>>
>>I get a(1)=b(1)=1; a(2)=1, b(2)=2 from S={1,2}; a(3)=2, b(3)=6 from
>>S={2,3,6}.
>>Of course in general the best S for a(n) will probably be different
>>from the best S for b(n), and won't be unique, either.
>>
>>Neil
>>
>>
>>
>
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