Sigma function σ1 (n ) up to n = 250 Prime-power factors In math , a colossally abundant number (also written as CA ) is a type of natural number that has to follow a special set of rules . CAs usually have a lot of divisors . To figure out whether or not a number is a CA, however, it has to follow an equation . For a number to be colossally abundant, ε has to be greater than 0. k a number greater than 1 and σ is the sum of every divisor that the number has.[1]
σ
(
n
)
n
1
+
ε
≥
σ
(
k
)
k
1
+
ε
{\displaystyle {\frac {\sigma (n)}{n^{1+\varepsilon ))}\geq {\frac {\sigma (k)}{k^{1+\varepsilon ))))
All colossally abundant numbers are also superabundant numbers , but not all superabundant numbers are colossal.
The first 15 colossally abundant numbers are 2 , 6 , 12 , 60 , 120 , 360, 2520, 5040, 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A004490 in the OEIS ). These are also the first 15 superior highly composite numbers.
Colossally abundant numbers were first learned about by Ramanujan . They were written about in his paper about highly composite numbers in 1915.[2] [3] [4]
In 1944, Leonidas Alaoglu and Paul Erdős expanded on what Ramanujan's wrote about and learned more about it.[5]
In the 1980s, Guy Robin showed[6] that the Riemann hypothesis is the same for whenever n is greater than 5040(γ is the Euler–Mascheroni constant ).
σ
(
n
)
<
e
γ
n
log
log
n
≈
1.781072418
⋅
n
log
log
n
{\displaystyle \sigma (n)<e^{\gamma }n\log \log n\approx 1.781072418\cdot n\log \log n\,}
This doesn't work for 27 different numbers (sequence A067698 in the OEIS ):
2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, 5040 Robin showed that if the Riemann hypothesis is true then n = 5040 is the last integer that doesn't work in this equation. This inequality is also known as Robin's inequality.
From 2001–2002 Lagarias[7] showed that Robin's inequality can be written another way. This inequality uses the harmonic numbers instead of logarithms and works for any CA that is bigger than 60.
σ
(
n
)
<
H
n
+
exp
(
H
n
)
log
(
H
n
)
{\displaystyle \sigma (n)<H_{n}+\exp(H_{n})\log(H_{n})}
The next inequality works for when n is equal to 1, 2, 3, 4, 6, 12, 24 or 60.
σ
(
n
)
<
exp
(
H
n
)
log
(
H
n
)
{\displaystyle \sigma (n)<\exp(H_{n})\log(H_{n})}
↑ K. Briggs, "Abundant Numbers and the Riemann Hypothesis", Experimental Mathematics 15:2 (2006), pp. 251–256, doi :10.1080/10586458.2006.10128957 .
↑ S. Ramanujan, "Highly Composite Numbers", Proc. London Math. Soc. 14 (1915), pp. 347–407}.
↑ S. Ramanujan, Collected papers , Chelsea, 1962.
↑ S. Ramanujan, "Highly composite numbers. Annotated and with a foreword by J.-L. Nicholas
and G. Robin", Ramanujan Journal 1 (1997), pp. 119–153.
↑ Alaoglu, L.; Erdős, P. (1944), "On highly composite and similar numbers" (PDF) , Transactions of the American Mathematical Society , 56 (3): 448–469, doi :10.2307/1990319 , JSTOR 1990319 , MR 0011087 .
↑ G. Robin, "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", Journal de Mathématiques Pures et Appliquées 63 (1984), pp. 187–213.
↑ J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis , American Mathematical Monthly 109 (2002), pp. 534–543.
Divisibility-based sets of integers
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