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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 $\ge 2n$. If $𝐩=\left({p}_{1},{p}_{1},\cdots ,{p}_{k}\right)$ is a vector of different primes and $𝐚=\left({a}_{1},{a}_{2},\cdots ,{a}_{k}\right)$ is a vector of nonnegative integers, then we write ${𝐩}^{𝐚}={p}_{1}^{{a}_{1}}{p}_{2}^{{a}_{2}}\cdots {p}_{k}^{{a}_{k}}$ and the vector $𝐩$ is called abundant, if some number of the form ${𝐩}^{𝐚}$ is abundant. The author shows that a necessary and sufficient condition for $𝐩$ to be abundant is: ${\prod }_{i}{p}_{i}/\left({p}_{i}-1\right)\ge 2$.

He proves the following theorem. Suppose that $𝐩=\left({p}_{1},{p}_{2},\cdots ,{p}_{k}\right)$ is a fixed vector of successive primes with ${p}_{k}<{p}_{1}^{r}<2{p}_{k}$ for some integer $r$ and $𝐩$ is abundant. Suppose that for each integer $\xi$ with $1<\xi <{p}_{1}$ an equation of the form $\xi {𝐩}^{𝐛}={𝐩}_{1}^{{𝐜}_{1}}+\cdots +{𝐩}_{j}^{{𝐜}_{j}}$ holds, where ${𝐩}^{𝐛}>1$ and ${𝐜}_{i}$ are distinct. Then every rational positive number $X$ of the form $A/{𝐩}^{𝐚}$ has an Egyption fraction representation $X=1/{X}_{1}+\cdots +1/{X}_{n}$ where ${X}_{i}$ distinct, of the form ${𝐩}_{i}^{𝐚}$. As an example he shows $𝐩=\left(3,5,7\right)$ 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, etc.