[seqfan] Re: Phil Scovis's problem

[Neil Sloane]

Charles, I agree a(n) >= A061799(n-1). Can you prove equality?

On Tue, Mar 5, 2013 at 1:35 PM, Charles Greathouse <
charles.greathouse at case.edu> wrote:

> I think a(n) is just A061799 with a different offset. b(n) seems more
> interesting (and harder).
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
>
> On Tue, Mar 5, 2013 at 1:14 PM, Neil Sloane <njasloane at gmail.com> 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
> >
> >
> > --
> > Dear Friends, I have now retired 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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> >
>
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-- 
Dear Friends, I have now retired 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