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On highly composite numbers. (English) Zbl 0061.07904
A (positive whole) number is called highly composite if it has more divisors than any smaller number, highly abundant if the sum of its divisors is greater than that for any smaller number, and superabundant if the sum of the reciprocals of its divisors is greater than that for any smaller number. The author uses A. E. Ingham’s theorem on the difference between consecutive primes [Q. J. Math., Oxf. Ser. 8, 255–266 (1937; Zbl 0017.38904)] to prove that if $$n$$ is highly composite, there is another highly composite number between $$n$$ and $$n+n(\log n)^{-c}$$, where $$c$$ is a certain positive constant.
Reviewer: P. T. Bateman

##### MSC:
 11A25 Arithmetic functions; related numbers; inversion formulas
##### Keywords:
highly composite numbers
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