## 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)

### MSC:

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