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**Zentralblatt MATH**

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

**Zbl.No: ** 151.03501

**Autor: ** Erdös, Pál

**Title: ** On some properties of prime factors of integers (In English)

**Source: ** Nagoya Math. J. 27, 617-623 (1966).

**Review: ** Let n = **prod**_{i = 1}^{\nu (n)} be the canonical decomposition of an integer n > 1. Define for 2 \leq j \leq \nu(n) **prod**_{i = 1}^{j-1} p_{i}^{\alpha i} = p_{j}^{\gammaj(n)} and set

**max**_{2 \leq j \leq \nu(n)} \gamma_{j}(n) = P(n). The author proves the following results:

(1) for almost all integers n (i. e. for all integers n but possibly a sequence of integers of density 0) one has

P(n) = (1+o(1)) log_{3} n / log_{4} n; (2) there is a continuous strictly increasing function \phi (c) with \phi (0) = 0, \phi (oo) = 1 such that for almost all integers n

{1 \over log_{2}n} **sum**_{\gammaj (n) \leq c} 1 ––> \phi (c); (3) the density of integers n for which **max**_{2 \leq j \leq \nu (n)} \gamma_{i}(n) < c/ log_{2} n is given by \psi (c), where \psi (c) is a continuuous strictly increasing function with \psi (0) = 0, \psi(oo) = 1. Here, log_{1} n = log n and log_{k}n = log (log_{k-1} n) for k = 2,3,4.

**Reviewer: ** S.Uchiyama

**Classif.: ** * 11N25 Distribution of integers with specified multiplicative constraints

**Index Words: ** number theory

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