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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 2n. If 𝐩=(p 1 ,p 1 ,,p k ) is a vector of different primes and 𝐚=(a 1 ,a 2 ,,a k ) is a vector of nonnegative integers, then we write 𝐩 𝐚 =p 1 a 1 p 2 a 2 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: i p i /(p i -1)2.

He proves the following theorem. Suppose that 𝐩=(p 1 ,p 2 ,,p k ) is a fixed vector of successive primes with p k <p 1 r <2p k for some integer r and 𝐩 is abundant. Suppose that for each integer ξ with 1<ξ<p 1 an equation of the form ξ𝐩 𝐛 =𝐩 1 𝐜 1 ++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 ++1/X n where X i distinct, of the form 𝐩 i 𝐚 . As an example he shows 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:
11D68Rational numbers as sums of fractions
11A25Arithmetic functions, etc.