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).