## 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 = prodi = 1\nu (n) be the canonical decomposition of an integer n > 1. Define for 2 \leq j \leq \nu(n)

prodi = 1j-1 pi\alpha i = pj\gammaj(n)

and set

max2 \leq j \leq \nu(n) \gammaj(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)) log3 n / log4 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 log2n} sum\gammaj (n) \leq c 1 ––> \phi (c);

(3) the density of integers n for which max2 \leq j \leq \nu (n) \gammai(n) < c/ log2 n is given by \psi (c), where \psi (c) is a continuuous strictly increasing function with \psi (0) = 0, \psi(oo) = 1. Here, log1 n = log n and logkn = log (logk-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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