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**Zentralblatt MATH**

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

**Zbl.No: ** 401.10068

**Autor: ** Erdös, Paul; Sarközy, A.

**Title: ** On products of integers. II. (In English)

**Source: ** Acta Sci. Math. 40, 243-259 (1978).

**Review: ** Let k,n be any positive integers. A = **{**a_{1},...,a_{n}**}** any finite, strictly increasing sequence of positive integers satisfying (*) a_{1} = 1, a_{2} = 2, ... a_{k} = k. Let us denote the number of integers which can be written in the form **prod**_{i = 1}^{n}a_{i}^{\epsiloni}(\epsilon_{1} = 0 or 1) or a_{i}a_{j} (1 \leq i, j \leq n), respectively by f(A,n,k) and g(A,n,k). Let us write F(n,k) = **max**_{A}f(A,n,k) and G(n,k) = **max**_{A}g(A,n,k), where the minima are extended over all sequence A satisfying (*) and |A| = n. The authors conjectured in an earlier paper [Studia Sci. Math. Hung. 9, 161-171 (1974; Zbl 304.10034)] that (1) G(n,k)/n > c_{1}. G(k,k)/k for every n \geq k, and furthermore, that for any \omega > o, k > k_{0}(\omega) and n \geq k, we have F(n,k) > n^{2}k^{\omega} or perhaps (2) n^{2}\exp**(**\frac{c_{2}k}{log k}**)** < F(n,k) < n^{2}\exp(c_{3}k/ log k) for large k and n \geq k. In this paper, the authors disprove (1) and prove a slightly weaker form of (2).

**Reviewer: ** D.Suryanarayana

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

11N13 Primes in progressions

**Keywords: ** products of integers; distribution in sequences

**Citations: ** Zbl.304.10034

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