## 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

number theory
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### References:

 [1] DOI: 10.4153/CJM-1959-021-x · Zbl 0092.04301 [2] Proc. Cambridge Phil. Soc 32 pp 532– (1936) [3] Bull. Amer. Math. Soc 52 pp 535– (1946) [4] J. London Math. Soc 9 pp 274– (1934) [5] Quart. J. Math. 48 pp 76– (1917) [6] Indigationes Math. 13 pp 50– (1945)
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