A065825 -- Sequences containing no 3-term arithmetic progressions -- Roth's problem

[Ed Pegg]

I added a 1953 ref to to 
  https://oeis.org/A065825  

This could maybe be called Roth's problem.

[jpallouche.math]

Yes and no if I may say: what about the papers of Salem and Spencer dated 1942 and 1950?

Salem, R.; Spencer, D. C.
On sets of integers which contain no three terms in arithmetical progression.
Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 561–563.
(https://pmc.ncbi.nlm.nih.gov/articles/PMC1078539/)

and

Salem, R.; Spencer, D. C. On sets which do not contain a given number of terms in arithmetical progression. 
Nieuw Arch. Wiskunde (2) 23 (1950), 133–143.


also see https://en.wikipedia.org/wiki/Salem%E2%80%93Spencer_set


best wishes
jean-paul

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[jpallouche.math]

Sorry for my many answers: my mailer became uncontrolled :(
jpa

[Ed Pegg]

Thanks, Neil.  
I wanted to see what these all looked like, so I collected all the solutions up to a(43)  and made a picture. 
An a(44) candidate, with a high value of 215, is (a(22), a(22) + 141) 

On Thu, Jul 30, 2026 at 11:04 AM jpallouche.math <jpallou...@gmail.com> wrote:

Sorry for my many answers: my mailer became uncontrolled :(
jpa

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[Ed Pegg]

Show all the solutions in a positional graphic.