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

[Zach DeStefano]

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
>
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