[seqfan] Phil Scovis's problem

[Neil Sloane]

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
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