[seqfan] Re: Largest subsets of {1, ..., n} such that no difference is ...

[Zach DeStefano]

It appears that (a) already has a sequence: A100719.
As for (b), (c), and (d), here are the first 40 values. They don't match
any existing sequence.
b: 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 6, ...
c: 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6,
7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, ...
d: 1, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,
4, 4, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 8, 8, 8, ...
I would be happy to provide b-files with more terms.
(c) empirically seems to be following a similar mundane pattern to the sum
case with maximal subsets generally just following the form: all 2k + 4 <=
n.

On Mon, May 15, 2023 at 5:52 PM Zach DeStefano <zachdestefano at gmail.com>
wrote:

> About two years ago, I had thought about a similar problem, namely that of
> maximal subsets of {1,...n} where SUMs of pairs avoid squares, determined
> lower and upper bounds on the size of the subset, and computed the maximum
> sized subsets up to n = 100. My results are here:
> http://www.mathmasterzach.com/Math/SquarelessSets.html. The results I got
> were all very close to the lower bound and would appreciate it if anyone
> could improve the upper bound.
>
> I had also thought about the primes, but the results there were less
> interesting (bet that's the only time you will hear that about the primes).
> In general, the optimal subsets with sums avoiding primes for the values of
> n I tested just consisted of all even numbers <= n.
>
> I can easily adapt my code to handle differences and contribute terms here.
>
> Also, is there a link to a recording of this talk online?
>
> - Zach
>
> On Mon, May 15, 2023 at 4:17 PM Neil Sloane <njasloane at gmail.com> wrote:
>
>> Dear Seqfans, A recent talk by Ben Green studies the density of, for
>> example, the largest subset of {1..n} such that no difference is:
>> (a) a square, (b) a prime - 1, (c) a prime, or (d) a prime + 1.
>>
>> So let's look at the actual sizes, not the density.
>>
>> I started off with (c), so let a(n) = max subset of {1..n} such that no
>> difference is a prime.  For n=1..11 I get
>> 1,2,2,2,2,2,2,2,3,3,4
>> which seems not to be in the OEIS.
>> The examples where a(n) increases are {1}, {1,2}, {1,5,9}, {1,2,10,11}.
>> Could someone check?
>>
>> And what about (a), (b), and (d)?
>>
>> Best regards
>> Neil
>>
>> Neil J. A. Sloane, Chairman, OEIS Foundation.
>> Also Visiting Scientist, Math. Dept., Rutgers University,
>> Email: njasloane at gmail.com
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>