[seqfan] Conflicting versions of "no 4-term AP"

[Neil Sloane]

Dear Seq Fans,
A correspondent, Lewis Chen, points out that the definitions
of A240555  (Earliest positive increasing sequence with no 4-term
arithmetic progressions.) and A005837 (Sequence contains no 4-term
arithmetic progression, based on a paper by J. L. Gerver and L. T. Ramsey,
Sets of integers with no long arithmetic progressions generated by the
greedy algorithm)  seem incompatible. The latter is both lexicographically
earlier than the former, and also grows much more slowly than it.

Both sequences have Mma programs from Tony Noe, as follows:

A240555:
t = {1, 2, 3}; Do[s = Table[Append[i, n], {i, Subsets[t, {3}]}]; If[!
MemberQ[Flatten[Table[Differences[i, 3], {i, s}]], 0], AppendTo[t, n]], {n,
4, 1000}]; t

A005837:
t = {1, 2, 3}; Do[s = Table[Append[i, n], {i, Subsets[t, {3}]}]; If[!
MemberQ[Table[Differences[i, 2], {i, s}], {0, 0}], AppendTo[t, n]], {n, 4,
200}]; t

Could one of our Mma experts explain the difference?

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com