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."