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Zbl 0781.11015
Friedman, Charles N.
Sums of divisors and Egyptian fractions.
(English)
[J] J. Number Theory 44, No.3, 328-339 (1993). ISSN 0022-314X; ISSN 1096-1658/e

The author discusses the presentation of rational numbers as a sum of Egyptian fractions, i.e. fractions of the form $1/X\sb i$, $X\sb i$ integers $>1$, and related problems. A number $n$ is called abundant, if the sum of all positive divisors of $n$ is $\geq 2n$. If ${\bold p}=(p\sb 1,p\sb 1,\dots,p\sb k)$ is a vector of different primes and ${\bold a}=(a\sb 1,a\sb 2,\dots,a\sb k)$ is a vector of nonnegative integers, then we write ${\bold p}\sp{\bold a}= p\sb 1\sp{a\sb 1} p\sb 2\sp{a\sb 2} \cdots p\sb k\sp{a\sb k}$ and the vector ${\bold p}$ is called abundant, if some number of the form ${\bold p}\sp{\bold a}$ is abundant. The author shows that a necessary and sufficient condition for ${\bold p}$ to be abundant is: $\prod\sb i p\sb i/(p\sb i- 1)\geq 2$.\par He proves the following theorem. Suppose that ${\bold p}=(p\sb 1,p\sb 2,\dots, p\sb k)$ is a fixed vector of successive primes with $p\sb k<p\sb 1\sp r<2p\sb k$ for some integer $r$ and ${\bold p}$ is abundant. Suppose that for each integer $\xi$ with $1<\xi<p\sb 1$ an equation of the form $\xi{\bold p}\sp{\bold b}={\bold p}\sb 1\sp{{\bold c}\sb 1}+ \cdots+ {\bold p}\sb j \sp{{\bold c}\sb j}$ holds, where ${\bold p}\sp{\bold b}>1$ and ${\bold c}\sb i$ are distinct. Then every rational positive number $X$ of the form $A/{\bold p}\sp{\bold a}$ has an Egyption fraction representation $X=1/X\sb 1+ \cdots+1/X\sb n$ where $X\sb i$ distinct, of the form ${\bold p}\sp{\bold a}\sb i$. As an example he shows ${\bold p}=(3,5,7)$ and $1=1/3+ 1/5+ 1/7+ 1/9+1/15+ 1/21+ 1/27+ 1/35+ 1/45+ 1/105+ 1/945$.
[T.Tonkov (Sofia)]
MSC 2000:
*11D68 Rational numbers as sums of fractions
11A25 Arithmetic functions, etc.

Keywords: semiperfect numbers; weird numbers; abundant numbers; rational numbers; sum of Egyptian fractions

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