Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 272.10003
Autor: Erdös, Paul
Title: Über die Zahlen der Form \sigma(n)-n und n-\phi(n). (On the numbers of the form \sigma(n)-n und n-\phi(n).) (In German)
Source: Elemente Math. 28, 83-86 (1973).
Review: In 1955, Sierpinski and the author discussed Euler's \phi-function and believed that the equation (1) n-\phi(n) = m is insoluble for an infinity of values of m. Although no proof of this belief is yet available, the author considers the analogous problem: (2) \sigma(n)-n = m is insoluble for an infinity of values of m. He proves the statement with the aid of two theorems. Theorem 1: The lower density of the numbers m for which (2) is insoluble is positive. Theorem 2: Given \epsilon > 0 there exists a k such that for all x > x0(\epsilon,k) the function A(k,x) of numbers n \ne prime p for which \sigma(n)-n \leq x, \sigma(n)-n \equiv 0 (mod Pk) is valid is smaller than \epsilon x/Pk where Pk denotes the product of the first k primes. He next proves a well-known Lemma: Let p be an arbitrary prime. The density of the numbers n with \sigma(n) \not\equiv 0(mod p) is 0.
Reviewer: S.M.Kerawala
Classif.: * 11A25 Arithmetic functions, etc.
11B83 Special sequences of integers and polynomials
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