: Mathematics: colossally abundant number

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colossally abundant number

In number theory, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in a particular, rigorous sense, has many divisors. Particularly, it is defined by a ratio between the sum of an integer's divisors and that integer raised to a power higher than one. For any such exponent, whichever integer has the highest ratio is a colossally abundant number. It is a stronger restriction than that of a superabundant number, but not strictly stronger than that of an abundant number. Formally, a number $n$ is said to be colossally abundant if there is an $ε > 0$ such that for all $k > 1$ ,
${\\displaystyle {\ rac {\\sigma (n)}{n^{1+\\varepsilon }}}\\geq {\ rac {\\sigma (k)}{k^{1+\\varepsilon }}}}$
where $σ$ denotes the sum-of-divisors function.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Colossally_abundant_number)

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