[seqfan] Re: Phil Scovis's problem

[Charles Greathouse]

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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