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On a question about sum-free sequences. (English) Zbl 0958.11023
Eine monoton wachsende Folge natürlicher Zahlen $(n_1, n_2,\dots, n_k,\dots)$ heißt Summen-frei (sum-free), wenn kein Glied der Folge Summe von verschiedenen Folgengliedern ist. Es werden als neue Resultate bewiesen: Es gibt eine Summen-freie Folge $(n_k)$ mit $n_{k+1}/n_k\to 1$ für $k\to \infty$ (Theorem 3) sowie: Für jedes $\delta> 0$ gibt es eine Summen-freie Folge $(n_k)$ mit $n_k\sim k^{3+\delta}$ (Theorem 4). Die Beweise verlaufen konstruktiv.
MSC:
11B83Special sequences of integers and polynomials
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References:
[1] Abbott, H. L.: On sum-free sequences. Acta arith. 48, 93-96 (1987) · Zbl 0619.10055
[2] Erdo?s, P.: Some remarks on number theory, III. Mat. lapok 13, 28-38 (1962) · Zbl 0123.25503
[3] P. Erdo?s, personal communication, 29 May 1996.
[4] Kuipers, L.; Niederreiter, H.: Uniform distribution of sequences. (1974) · Zbl 0281.10001
[5] Levine, E.: An extremal result for sum-free sequences. J. number theory 12, 251-257 (1980) · Zbl 0431.10035
[6] Levine, E.; O’sullivan, J.: An upper estimate for the reciprocal sum of a sum-free sequence. Acta arith. 34, 9-24 (1977) · Zbl 0335.10053