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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 469.10036

**Autor: ** Erdös, Paul

**Title: ** Some extremal problems on divisibility properties of sequences of integers. (In English)

**Source: ** Fibonacci Q. 19, 208-213 (1981).

**Review: ** The author studies various properties of subsets A = **{**a_{1} < a_{2} < ... < a_{k} \leq n**}** of integers from 1 to n. The sequence A is said to have the property P_{r}(n) if no a_{i} divides the product of r of the others. A is said to have the property P(n) if no a_{i} divides the product of the others. It is said to have the property Q(n) if the products a_{i} a_{j} are all distinct. To his results in [Izv. Nauk. Inst. Mat. Mekh. Univ. Tomsk 2, 74-82 (1938; Zbl 020.00504)] he adds further results in this paper. to state them he defines some arithmetical functions. Let S_{n} denote all the integers from 1 to n. Let f_{r}(n) denote the smallest integer such that S_{n} can be decomposed into g(n) sets each with the property Q(n). The author proves the results 2n^{ ½} > f_{r}(n) >> n^{ ½}/ log n and 2n^{ ½} > g(n) >> n^{1/3}/ log n. Next he proves the results that for every \epsilon > 0, there holds n^{1-1/r} >> f_{r}(n) >> \epsilon^{n^{1-(1/r)-\epsilon}}. Another result about the divisor properties is that if A has the property that the product of any two a_{i} is a multiple of all the others, then log(**max** k) is asymptotic to **(**\frac{2 log2}{3}**)**\frac{log n}{log log n} as n ––> oo. I wish to quote on more result. Let F(n) be the smallest integer for which S_{n} can be decomposed into F(n) sets **{**A_{i}**}**1 \leq i \leq F(n) each having the property P. Then F(n) = nExp((-c+o(1))(log n log log n)^{ ½}). The author discusses many other deep results about sets A with a_{i}+a_{j} all distinct and so on. As usual the paper is full of interesting problems and partial solutions (sometimes complete solutions) and i would request the readers to consult the paper for more details.

**Reviewer: ** K.Ramachandra

**Classif.: ** * 11B83 Special sequences of integers and polynomials

11B05 Topology etc. of sets of numbers

11B13 Additive bases

**Keywords: ** extremal problems; divisibility properties; sequences of integers

**Citations: ** Zbl.020.005

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