Colossally Abundant Number
A colossally abundant number is a positive integer 
for which there is a positive exponent 
such that
for all 
. All colossally abundant numbers are superabundant numbers.
The first few are 2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800, 160626866400, ... (OEIS A004490). The following table lists the colossally abundant numbers up to 
, as given by Alaoglu and Erdős (1944).
 |
factorization of  |
 |
| 2 |
2 |
1.500 |
| 6 |
 |
2.000 |
| 12 |
 |
2.333 |
| 60 |
 |
2.800 |
| 120 |
 |
3.000 |
| 360 |
 |
3.250 |
| 2520 |
 |
3.714 |
| 5040 |
 |
3.838 |
| 55440 |
 |
4.187 |
| 720720 |
 |
4.509 |
| 1441440 |
 |
4.581 |
| 4324320 |
 |
4.699 |
| 21621600 |
 |
4.855 |
| 367567200 |
 |
5.141 |
| 6983776800 |
 |
5.412 |
| 160626866400 |
 |
5.647 |
| 321253732800 |
 |
5.692 |
| 9316358251200 |
 |
5.888 |
| 288807105787200 |
 |
6.078 |
| 2021649740510400 |
 |
6.187 |
| 6064949221531200 |
 |
6.238 |
| 224403121196654400 |
 |
6.407 |
The first 15 elements of this sequence agree with those of the superior highly composite numbers (OEIS A002201).
The 
th colossally abundant number 
has the form 
, where 
,
, ... is a sequence of non-distinct prime numbers. The first few of these primes are 2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 23, ... (OEIS A073751).
REFERENCES:
Alaoglu, L. and Erdős, P. "On Highly Composite and Similar Numbers." Trans. Amer. Math. Soc. 56, 448-469, 1944.
Lagarias, J. C. "An Elementary Problem Equivalent to the Riemann Hypothesis." Amer. Math. Monthly 109, 534-543, 2002.
Sloane, N. J. A. Sequences A002201, A004490 and A073751 in "The On-Line Encyclopedia of Integer Sequences."
