Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 012.01101
Autor: Erdös, Paul
Title: Note on consecutive abundant numbers. (In English)
Source: J. London Math. Soc. 10, 128-131 (1935).
Review: Continuing his work on abundant numbers (Zbl 010.10303; Zbl 010.39103), the author proves that there are two absolute constants c1, c2 such that for all large n there are at least c1 log log log n but not more than c2 log log log n consecutive abundant numbers less than n. The first result is obtained by taking a1 = 2· 3, a2 = 5· 7...p1, a3 = p2...p3,..., where p1 is the least prime making a2 abundant, p2 the next prime, p3 the least prime making a3 abundant, and so on, and solving the congruences x\equiv r-1 (mod ar), r = 1,2,...,\nu. The second result is proved by considering those numbers b1,...,b2, of a set of k consecutive abundant numbers, which are not divisible by any prime less than a particular fixed prime q. We have 2z \leq prodi = 1z {\sigma (bi) \over bi} < prod \Sb{p > q}
{p | b1,...bz}\endSb ({p \over p-1})[ k/p ]+1, and
z > k prodp < q (1- 1/p ) - 2q, the latter by the sieve of Eratosthenes. From these inequalities the upper bound for k is deduced.
Reviewer: Davenport (Cambridge)
Classif.: * 11A25 Arithmetic functions, etc.
Index Words: Number theory
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