Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 312.10003
Autor: Erdös, Paul
Title: On abundant-like numbers. (In English)
Source: Can. Math. Bull. 17, 599-602 (1974).
Review: Let 2-p1 < p2 < ... be the sequence of primes. Denote by nk(c) the smallest integer for which pk is the smallest prime divisor of n(c)k and \sigma (nk(c) \geq cnk(c) where \sigma (n) denotes the sum of the divisors of n. From the reviewer's solution of a problem proposed by the author [Canadian math. Bull. 16, 144 (1973)] it follows that there are only a finite number of squarefree integers which are nk(c)'s for some c \geq 2 (maybe only the integer 6). The author now proves that nk(2) is cubefree for all sufficiently large k. The proof depends on a method developed by Ramanujan. The situation for 1 < c < 2 is much more complicated. It is shown e.g. that the sets of numbers c for which nk(c) is infinitely often squarefree resp. not squarefree are both dense in (1,2).
Reviewer: J.H.van Lint
Classif.: * 11A05 Multiplicative structure of the integers
11A25 Arithmetic functions, etc.
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