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

Rob Pratt Rob.Pratt at sas.com
Mon May 15 23:28:45 CEST 2023

```I confirmed your values for (c).

(a): https://oeis.org/A100719
(b): https://oeis.org/A131849
(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 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 11 12 12 12 12 12 12 12

-----Original Message-----
From: SeqFan <seqfan-bounces at list.seqfan.eu> On Behalf Of Neil Sloane
Sent: Monday, May 15, 2023 4:17 PM
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
Subject: [seqfan] Largest subsets of {1, ..., n} such that no difference is ...

EXTERNAL

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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