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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### Citations:

Zbl 0017.38904
Full Text:
DOI

### Online Encyclopedia of Integer Sequences:

Highly composite numbers, definition (1): numbers n where d(n), the number of divisors of n (A000005), increases to a record.Ramanujan’s largely composite numbers, defined to be numbers m such that d(n) >= d(k) for k = 1 to m-1.

Highly composite numbers (A002182) whose following highly composite number is at least 3/2 times greater.

Number of Ramanujan’s largely composite numbers having prime(n) as the greatest prime divisor.

Ramanujan’s largely composite numbers n (A067128) which are not divisible by all the primes < p, where p is the greatest prime divisor of n.

The smallest term of A273379 having n primes between two consecutive prime divisors.

The number of highly composite numbers between the factorials n! and (n+1)!.

The number of highly composite numbers between 2^n and 2^(n+1).