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**Zentralblatt MATH**

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

**Zbl.No: ** 393.10046

**Autor: ** Erdös, Paul

**Title: ** Some unconventional problems in number theory. (In English)

**Source: ** Acta Math. Acad. Sci. Hung. 33, 71-80 (1979).

**Review: ** This fascinating miscellany covers a great many topics and the best this reviewer can do is attempt to give a taste of the paper. As stated by the author the paper mostly deals with arithmetic functions, primes, divisors, sieve processes and consecutive integers. Let f be an arithmetic function. The integer n is called a barrier for f if m+f(m) \leq n for every m < n. For n = **prod** p_{i}^{\alphai} put d_{0}(n) = **prod**\alpha_{i}. Then d_{0}(n) has infinitely many barriers, that is there are infinitaly many n such that m+d_{0}(m) \leq n for every m < n. In fact the density of integers n which are barriers for d_{0} is positive. An outline of the proof of this Theorem is given including the following observation. Let \epsilon > 0, k be a sufficiently large integer and A be a multiple of p_{1},p_{2},...,p_{k}. Then the density of integers t for which d_{0}(tA^{k}-i) > i, for some i with l \leq i \leq k, is less than ½ \epsilon. Among others the following problem is discussed. Let p(m) denote the least prime factor of m and put F(n) = **max****{**m+p(m): l \leq m < n, m composite**}**. Is it true that F(n) \leq n for infinitely many n?

**Reviewer: ** E.M.Horadam

**Classif.: ** * 11N37 Asymptotic results on arithmetic functions

11-02 Research monographs (number theory)

11A25 Arithmetic functions, etc.

00A07 Problem books

**Keywords: ** arithmetic function; consecutive integers; problems; primes; divisors sieve processes

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