Colossally abundant number

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Divisibility-based
sets of integers
Form of factorization:
Prime number
Composite number
Powerful number
Square-free number
Achilles number
Constrained divisor sums:
Perfect number
Almost perfect number
Quasiperfect number
Multiply perfect number
Hyperperfect number
Superperfect number
Unitary perfect number
Semiperfect number
Primitive semiperfect number
Practical number
Numbers with many divisors:
Abundant number
Highly abundant number
Superabundant number
Colossally abundant number
Highly composite number
Superior highly composite number
Other:
Deficient number
Weird number
Amicable number
Friendly number
Sociable number
Solitary number
Sublime number
Harmonic divisor number
Frugal number
Equidigital number
Extravagant number
See also:
Divisor function
Divisor
Prime factor
Factorization

In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a certain kind of natural number. Formally, a number n is colossally abundant if and only if there is an ε > 0 such that for all k > 1,

\frac{\sigma(n)}{n^{1+\varepsilon}}\geq\frac{\sigma(k)}{k^{1+\varepsilon}}

where σ denotes the divisor function. The first few colossally abundant numbers are 2, 6, 12, 60, 120, 360, 2520, 5040, ... (sequence A004490 in OEIS); all colossally abundant numbers are also superabundant numbers, but the converse is not true.

Contents

Properties

All colossally abundant numbers are Harshad numbers.

Relation to the Riemann hypothesis

If the Riemann hypothesis is false, a colossally abundant number will be a counterexample. In particular, the RH is equivalent to the assertion that the following inequality is true for n > 5040:

\sigma(n)<\exp(\gamma) \cdot n \log\log n

where γ is the Euler–Mascheroni constant.

This result is due to Robin[1].

Lagarias[2] and Smith[3] discuss this and similar formulations of the RH.

References

  1. ^ G. Robin, "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", Journal de Mathématiques Pures et Appliquées 63 (1984), pp. 187-213.
  2. ^ J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, American Mathematical Monthly 109 (2002), pp. 534-543.
  3. ^ Warren D. Smith, A "good" problem equivalent to the Riemann hypothesis, 2005

External links

Retrieved from "http://en.wikipedia.org/wiki/Colossally_abundant_number"

This article is from Wikipedia. All text is available under the terms of the GNU Free Documentation License.

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