Sets of integers with nonlong arithmetic progressions generated by the greedy algorithm

Authors:
Joseph L. Gerver and L. Thomas Ramsey

Journal:
Math. Comp. **33** (1979), 1353-1359

MSC:
Primary 10L10

DOI:
https://doi.org/10.1090/S0025-5718-1979-0537982-0

MathSciNet review:
537982

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Abstract: Let be the set of positive integers containing no arithmetic progression of *k* terms, generated by the greedy algorithm. A heuristic formula, supported by computational evidence, is derived for the asymptotic density of in the case where *k* is composite. This formula, with a couple of additional assumptions, is shown to imply that the greedy algorithm would not maximize over all *S* with no arithmetic progression of *k* terms. Finally it is proved, without relying on any conjecture, that for all , the number of elements of which are less than *n* is greater than for sufficiently large *n*.

**[1]**P. ERDÖS & P. TURAN, "On certain sequences of integers,"*J. London Math. Soc.*, v. 11, 1936, pp. 261-264.**[2]**J. GERVER, "The sum of the reciprocals of a set of integers with no arithmetic progression of*k*terms,"*Proc. Amer. Math. Soc.*, v. 62, 1977, pp. 211-214. MR**0439796 (55:12678)**

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DOI:
https://doi.org/10.1090/S0025-5718-1979-0537982-0

Article copyright:
© Copyright 1979
American Mathematical Society