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,

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.

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:

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

External links

• 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