[seqfan] Re: Phil Scovis's problem

[Neil Sloane]

Here's a summary of these three sequences so far:
Phil's original problem: A213918 [min-min]
The sequence I called a(n): A061799 [min-min]
The sequence I called b(n): A214799 [min-max] (added today)
Thanks to everyone who contributed.

For completeness, there could be a fourth sequence, the "b(n)"
or "min-max" version of A213918, in case someone would like to
calculate it. The definition would be:

Let S be a set of n positive integers s_1, s_2, ..., s_n such that for i !=
j,  |s_i - s_j| = GCD(s_i, s_j), and let  M(S) denote the largest element
of S; a(n) is the minimal value of M(S) over all choices for S.

I think if you look at the Pink Box comments in the History tab of A213909
(which was the original submission of A213918) you will see it is the
"min-max" version that Phil originally submitted.

Of course the examples in A213918 (and Phil's original calculations
in the above-mentioned History) provide upper bounds on the new sequence.

Neil

On Wed, Mar 6, 2013 at 12:03 PM, Giovanni Resta <g.resta at iit.cnr.it> wrote:

> On 03/06/2013 01:03 AM, israel at math.ubc.ca wrote:
>
>> 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.
>>
>
> I confirm b(9) = 960, with {504, 640, 672, 720, 756, 810, 896, 945, 960}
> and also b(10) = 1920 with {1008, 1215, 1280, 1344, 1440, 1512, 1620,
> 1792, 1890, 1920}.
>
> Giovanni Resta
>
>
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Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
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