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**Zentralblatt MATH**

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

**Zbl.No: ** 393.10047

**Autor: ** Erdös, Paul; Hall, R.R.

**Title: ** On some unconventional problems on the divisors of integers. (In English)

**Source: ** J. Aust. Math. Soc., Ser. A 25, 479-485 (1978).

**Review: ** The authors prove several theorems concerning the divisors of an integer. Let \tau(n) be the number of divisors of n, and let d_{1},...,d_{\tau} be all divisors of n, ordered so that 1 = d_{1} < d_ < ... < d_{\tau} = n. Let f(n) = card**{**i: (d_{i},d_{i+1}) = 1**}**. Next, let \tau_{k}(n) be the number of divisors of n of the form d = t(t+1)...(t+k-1), and let t_{k}(n) = **max****{**t \geq 1: n| t(t+1)...(t+k-1)**}**. The following results are proven:

Theorem 1. For every \epsilon > 0 and x > x_{0}(\epsilon) **max** f(m) > \exp((log log x)^{2-\epsilon}).

Theorem 2. for each k \geq 2 and every fixed A < e^{1/k} we have \tau_{k}(n) > (log n)^{A} infinitely often.

Theorem 3.

^{1}/_{x} **sum**_{n \leq x}t_{2}(n) << x\frac{log log log x}{log log x}

The last proven result involves the divisors of two integers. We say that two integers m and n interlock if every pair of divisors of n are separated by a divisor of m, and conversely (except for 1 and the smallest prime factor ofmn). An integer n is said to be separable if there exists an integer m such that m and n interlock, and let A(x) be the number of separable n \leq x. It is proven: Theorem 4. For every fixed c' > 0 and sufficiently large x we have

A(x) > c'x/ log log x.

**Reviewer: ** G.Kolesnik

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

**Keywords: ** asymptotic results; divisor function

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