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,

σ

(

n

)

n

1

+

ε

≥

σ

(

k

)

k

1

+

ε

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

where σ denotes the sum-of-divisors function. That is,

n

{\displaystyle n}

attains the maximum value of

σ

(

k

)

k

1

+

ε

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

.

The first 15 colossally abundant numbers, 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A004490 in the OEIS) are also the first 15 superior highly composite numbers, but neither set is a subset of the other.

History

Colossally abundant numbers were first studied by Ramanujan and his findings were intended to be included in his 1915 paper on highly composite numbers. Unfortunately, the publisher of the journal to which Ramanujan submitted his work, the London Mathematical Society, was in financial difficulties at the time and Ramanujan agreed to remove aspects of the work to reduce the cost of printing. His findings were mostly conditional on the Riemann hypothesis and with this assumption he found upper and lower bounds for the size of colossally abundant numbers and proved that what would come to be known as Robin's inequality (see below) holds for all sufficiently large values of n.