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 d1,...,d\tau be all divisors of n, ordered so that 1 = d1 < d_ < ... < d\tau = n. Let f(n) = card{i: (di,di+1) = 1}. Next, let \tauk(n) be the number of divisors of n of the form d = t(t+1)...(t+k-1), and let tk(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 > x0(\epsilon)

max f(m) > \exp((log log x)2-\epsilon).


Theorem 2. for each k \geq 2 and every fixed A < e1/k we have \tauk(n) > (log n)A infinitely often.
Theorem 3.

1/x sumn \leq xt2(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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