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On some properties of prime factors of integers. (English) Zbl 0151.03501

Let \(n = \prod_{i=1}^{\nu (n)}\) be the canonical decomposition of an integer \(n>1\). Define for \(2\leq j \leq \nu(n)\) \[ \prod_{i=1}^{j-1} p_i^{\alpha i}=p_j^{\gamma_j(n)} \] 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 \(\varphi (c)\) with \(\varphi (0)=0, \varphi (\infty)=1\) such that for almost all integers \(n\) \[ {1 \over \log_2n} \sum_{\gamma_j (n) \leq c} 1 \to \varphi (c); \] (3) the density of integers \(n\) for which \(\min_{2 \leq j \leq \nu (n)} \gamma_i(n) < c/\log_2 n\) is given by \(\psi (c)\), where \(\psi (c)\) is a continuous strictly increasing function with \(\psi (0)=0, \psi(\infty)=1\). Here, \(\log_1 n = \log n\) and \(\log_kn = \log (\log_{k-1} n)\) for \(k=2,3,4\).
Reviewer: S.Uchiyama

MSC:

11N25 Distribution of integers with specified multiplicative constraints

Keywords:

number theory
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[1] DOI: 10.4153/CJM-1959-021-x · Zbl 0092.04301
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