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On highly composite and similar numbers. (English) Zbl 0061.07903

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 obtain various results about these classes of numbers.
Reviewer: P. T. Bateman

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas

Citations:

Zbl 0017.38904
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Online Encyclopedia of Integer Sequences:

Highly abundant numbers: numbers k such that sigma(k) > sigma(m) for all m < k.
Superabundant [or super-abundant] numbers: n such that sigma(n)/n > sigma(m)/m for all m < n, sigma(n) being A000203(n), the sum of the divisors of n.
Colossally abundant numbers: m for which there is a positive exponent epsilon such that sigma(m)/m^{1 + epsilon} >= sigma(k)/k^{1 + epsilon} for all k > 1, so that m attains the maximum value of sigma(m)/m^{1 + epsilon}.
Prime numbers that when multiplied in order yield the sequence of colossally abundant numbers A004490.
Highly abundant numbers that are not superabundant, i.e., the complement of A004394 w.r.t. A002093.
Highly abundant numbers with an odd divisor sum.
Highly abundant numbers that are not products of consecutive primes with nonincreasing exponents, i.e., that are not of the form n=2^{e_2} * 3^{e_3} * ...* p^{e_p}, with e_2>=e_3>=...>=e_p.
Highly abundant numbers (A002093) that are not Harshad numbers (A005349).
Superabundant numbers (A004394) that are not highly composite (A002182).
Highly composite numbers that are not highly abundant numbers.
Highly abundant numbers (A002093) whose largest prime factor has power greater than 1.
Integer part of sigma(m)/phi(m) for colossally abundant numbers m.
Least highly abundant number with n distinct prime factors.
Integer part of sigma(m)*phi(m)/m for colossally abundant numbers m.
The number of primorials that neither exceed nor divide the n-th colossally abundant number.
Largest highly abundant number with n distinct prime factors
Numbers k such that there is no prime p and index j < k such that A002182(k) = p * A002182(j).
Highly abundant numbers that are powerful.
Numbers k such that there is no prime p and index j > k such that A002182(j) = p * A002182(k).