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
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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.

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[edit] Properties

All colossally abundant numbers are Harshad numbers.

[edit] 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.

[edit] 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

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