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# 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).

Abundant Number, Superabundant Number, Superior Highly Composite Number

This entry contributed by David Terr

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## 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."

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Colossally Abundant Number

## Cite this as:

Terr, David. "Colossally Abundant Number." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ColossallyAbundantNumber.html