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**Zentralblatt MATH**

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

**Zbl.No: ** 631.10029

**Autor: ** Erdös, Paul; Pomerance, Carl; Sárközy, András

**Title: ** On locally repeated values of certain arithmetic functions. III. (In English)

**Source: ** Proc. Am. Math. Soc. 101, 1-7 (1987).

**Review: ** The reviewer showed [Mathematika 31, 141-149 (1984; Zbl 529.10040)] that the number of n \leq x with d(n) = d(n+1) is at least of order x(log x)^{-7}. It is conjectured that the true order of magnitude is x(log log x)^{-} = f(x), say. The principal result of this paper is that the number of solutions is O(f(x)). A similar result for the equation \nu(n) = \nu(n+1) is also given, where \nu(n) is the number of distinct prime factors of n. As far as lower bounds are concerned, it was shown in the second paper of this series [Acta Math. Hung. 49, 251-259 (1987; Zbl 609.10034)] that the inequality |\nu(n)-\nu(n+1)| \leq 3 has >> f(x) solutions n \leq x.

The paper uses elementary methods, employing a simple sieve result. The article concludes with some results about the equation n+\nu(n) = m+\nu(m), and related topics.

**Reviewer: ** D.R.Heath-Brown

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

11N05 Distribution of primes

**Keywords: ** divisor function; values at consecutive integers; upper bounds; number of distinct prime factors

**Citations: ** Zbl 574.10012; Zbl 529.10040; Zbl 609.10034

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