Sets of integers with nonlong arithmetic progressions generated by the greedy algorithm
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- by Joseph L. Gerver and L. Thomas Ramsey PDF
- Math. Comp. 33 (1979), 1353-1359 Request permission
Abstract:
Let ${S_k}$ 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 ${S_k}$ 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 ${\Sigma _{n \in S}}1/n$ over all S with no arithmetic progression of k terms. Finally it is proved, without relying on any conjecture, that for all $\varepsilon > 0$, the number of elements of ${S_k}$ which are less than n is greater than $(1 - \varepsilon )\sqrt {2n}$ for sufficiently large n.References
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P. ERDÖS & P. TURAN, "On certain sequences of integers," J. London Math. Soc., v. 11, 1936, pp. 261-264.
- Joseph L. Gerver, The sum of the reciprocals of a set of integers with no arithmetic progression of $k$ terms, Proc. Amer. Math. Soc. 62 (1977), no. 2, 211–214. MR 439796, DOI 10.1090/S0002-9939-1977-0439796-9
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp. 33 (1979), 1353-1359
- MSC: Primary 10L10
- DOI: https://doi.org/10.1090/S0025-5718-1979-0537982-0
- MathSciNet review: 537982