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Colossally Abundant Number

A colossally abundant number is a positive integer n>1 for which there is a positive exponent epsilon such that

(sigma(n))/(n^(1+epsilon))>=(sigma(k))/(k^(1+epsilon))

for every integer k>1, where sigma is the divisor function. 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 10^(18), as given by Alaoglu and Erdős (1944), with sigma(n)/n rounded to three decimal places.

nfactorization of nsigma(n)/n
221.500
62·32.000
122^2·32.333
602^2·3·52.800
1202^3·3·53.000
3602^3·3^2·53.250
25202^3·3^2·5·73.714
50402^4·3^2·5·73.838
554402^4·3^2·5·7·114.187
7207202^4·3^2·5·7·11·134.509
14414402^5·3^2·5·7·11·134.582
43243202^5·3^3·5·7·11·134.699
216216002^5·3^3·5^2·7·11·134.856
3675672002^5·3^3·5^2·7·11·13·175.142
69837768002^5·3^3·5^2·7·11·13·17·195.412
1606268664002^5·3^3·5^2·7·11·13·17·19·235.648
3212537328002^6·3^3·5^2·7·11·13·17·19·235.692
93163582512002^6·3^3·5^2·7·11·13·17·19·23·295.889
2888071057872002^6·3^3·5^2·7·11·13·17·19·23·29·316.079
20216497405104002^6·3^3·5^2·7^2·11·13·17·19·23·29·316.187
60649492215312002^6·3^4·5^2·7^2·11·13·17·19·23·29·316.239
2244031211966544002^6·3^4·5^2·7^2·11·13·17·19·23·29·31·376.407

The first 15 elements of this sequence agree with those of the superior highly composite numbers (OEIS A002201).

Let c(n) be the nth colossally abundant number and set c(0)=1. The quotients p_n=c(n)/c(n-1) give c(n)=p_1p_2...p_n for n>=1. It is conjectured that every such quotient is a prime number (Alaoglu and Erdős 1944, p. 455). The initial quotients are 2, 3, 2, 5, 2, 3, 7, 2, 11, 13, 2, 3, 5, 17, 19, 23, ... (OEIS A073751).

See also

Abundant Number, Superabundant Number, Superior Highly Composite Number

Portions of 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."

Referenced on Wolfram|Alpha

Colossally Abundant Number

Cite this as:

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

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