Sums of divisors and Egyptian fractions. (English) Zbl 0781.11015

The author discusses the presentation of rational numbers as a sum of Egyptian fractions, i.e. fractions of the form \(1/X_ i\), \(X_ i\) integers \(>1\), and related problems. A number \(n\) is called abundant, if the sum of all positive divisors of \(n\) is \(\geq 2n\). If \({\mathbf p}=(p_ 1,p_ 1,\dots,p_ k)\) is a vector of different primes and \({\mathbf a}=(a_ 1,a_ 2,\dots,a_ k)\) is a vector of nonnegative integers, then we write \({\mathbf p}^{\mathbf a}= p_ 1^{a_ 1} p_ 2^{a_ 2} \cdots p_ k^{a_ k}\) and the vector \({\mathbf p}\) is called abundant, if some number of the form \({\mathbf p}^{\mathbf a}\) is abundant. The author shows that a necessary and sufficient condition for \({\mathbf p}\) to be abundant is: \(\prod_ i p_ i/(p_ i- 1)\geq 2\).
He proves the following theorem. Suppose that \({\mathbf p}=(p_ 1,p_ 2,\dots, p_ k)\) is a fixed vector of successive primes with \(p_ k<p_ 1^ r<2p_ k\) for some integer \(r\) and \({\mathbf p}\) is abundant. Suppose that for each integer \(\xi\) with \(1<\xi<p_ 1\) an equation of the form \(\xi{\mathbf p}^{\mathbf b}={\mathbf p}_ 1^{{\mathbf c}_ 1}+ \cdots+ {\mathbf p}_ j ^{{\mathbf c}_ j}\) holds, where \({\mathbf p}^{\mathbf b}>1\) and \({\mathbf c}_ i\) are distinct. Then every rational positive number \(X\) of the form \(A/{\mathbf p}^{\mathbf a}\) has an Egyption fraction representation \(X=1/X_ 1+ \cdots+1/X_ n\) where \(X_ i\) distinct, of the form \({\mathbf p}^{\mathbf a}_ i\). As an example he shows \({\mathbf p}=(3,5,7)\) and \(1=1/3+ 1/5+ 1/7+ 1/9+1/15+ 1/21+ 1/27+ 1/35+ 1/45+ 1/105+ 1/945\).
Reviewer: T.Tonkov (Sofia)


11D68 Rational numbers as sums of fractions
11A25 Arithmetic functions; related numbers; inversion formulas
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