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

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

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

Properties

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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 equivilent to the assertion that the sigma, the sum of the divisors of n, follows this constraint for n >= 5041:

This result is due to RobinG. Robin, "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", Journal de Mathématiques Pures et Appliquées 63 (1984), pp. 187-213.

LagariasJ. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, American Mathematical Monthly 109 (2002), pp. 534-543. and SmithWarren D. Smith, A "good" problem equivalent to the Riemann hypothesis, 2005 discuss this and similar formulations of the RH.

See also

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• Superabundant number
• Highly abundant number
• Abundant number
• Deficient number

External links

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• Keith Briggs on colossally abundant numbers and the Riemann hypothesis
• MathWorld entry
• Notes on the Riemann hypothesis and abundant numbers
• More on Robin's formulation of the RH

Integer sequences