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**Zentralblatt MATH**

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

**Zbl.No: ** 020.00504

**Autor: ** Erdös, Paul

**Title: ** On sequences of integers no one of which divides the product of two others and on some related problems. (In English)

**Source: ** Mitteil. Forsch.-Inst. Math. Mech. Univ. Tomsk 2, 74-82 (1938).

**Review: ** The author defines an A sequence of integers as a sequence such that no member divides the product of any two other members. The number of integers less than n belonging to such a sequence is less than \pi (n)+O **(**{n^ ^{1}/_{2} \over log n} **)**^{2}. The number of integers less than n and belonging to a sequence such that the product of any two members is different from any other such product is less than \pi (n)+O (n^ ^{1}/_{2}). The error term in the latter formula cannot be better than O(n^{3/4} (log n)^{-3/2}). It follows that, if p_{1} < p_{2} < ··· p_{z} \leq n is an arbitrary sequence of primes such that z > (c_{1} n log log n) (log n)^{-2}, where c_{1} is a sufficiently large constant, then the products (p_{i}-1) (p_{j}-1) cannot all be different.

**Reviewer: ** Wright (Aberdeen)

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

**Index Words: ** Number theory

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