Superabundant number

In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. A natural number n is called superabundant precisely when, for all m < n

{\frac {\sigma (m)}{m}}<{\frac {\sigma (n)}{n}}

where σ denotes the sum-of-divisors function (i.e., the sum of all positive divisors of n, including n itself). The first few superabundant numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ... (sequence A004394 in the OEIS). For example, the number 5 is not a superabundant number because for 1, 2, 3, 4, and 5, the sigma is 1, 3, 4, 7, 6, and 7/4 > 6/5.

Superabundant numbers were defined by Leonidas Alaoglu and Paul Erdős (1944). Unknown to Alaoglu and Erdős, about 30 pages of Ramanujan's 1915 paper "Highly Composite Numbers" were suppressed. Those pages were finally published in The Ramanujan Journal 1 (1997), 119–153. In section 59 of that paper, Ramanujan defines generalized highly composite numbers, which include the superabundant numbers.

Properties

Leonidas Alaoglu and Paul Erdős (1944) proved that if n is superabundant, then there exist a k and a₁, a₂, ..., a_k such that

n=\prod _{{i=1}}^{k}(p_{i})^{{a_{i}}}

where p_i is the i-th prime number, and

a_{1}\geq a_{2}\geq \dotsb \geq a_{k}\geq 1.

That is, they proved that if n is superabundant, the prime decomposition of n has non-increasing exponents (the exponent of a larger prime is never more than that a smaller prime) and that all primes up to $p_{k}$ are factors of n. Then in particular any superabundant number is an even integer, and it is a multiple of the k-th primorial $p_{k}\#.$

In fact, the last exponent a_k is equal to 1 except when n is 4 or 36.

Superabundant numbers are closely related to highly composite numbers. Not all superabundant numbers are highly composite numbers. In fact, only 449 superabundant and highly composite numbers are the same. For instance, 7560 is highly composite but not superabundant.

Alaoglu and Erdős observed that all superabundant numbers are highly abundant.

Not all superabundant numbers are Harshad numbers. The first exception is the 105th SA number, 149602080797769600. The digit sum is 81, but 81 does not divide evenly into this SA number.

Superabundant numbers are also of interest in connection with the Riemann hypothesis, and with Robin's theorem that the Riemann hypothesis is equivalent to the statement that

{\frac {\sigma (n)}{e^{\gamma }n\log \log n}}<1

for all n greater than the largest known exception, the superabundant number 5040. If this inequality has a larger counterexample, proving the Riemann hypothesis to be false, the smallest such counterexample must be a superabundant number (Akbary & Friggstad 2009).

References

Akbary, Amir; Friggstad, Zachary (2009), "Superabundant numbers and the Riemann hypothesis", American Mathematical Monthly, 116 (3): 273–275, doi:10.4169/193009709X470128 .
Alaoglu, Leonidas; Erdős, Paul (1944), "On highly composite and similar numbers", Transactions of the American Mathematical Society, American Mathematical Society, 56 (3): 448–469, doi:10.2307/1990319, JSTOR 1990319 .

External links

MathWorld: Superabundant number

Divisibility-based sets of integers

Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number

Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual

Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas

With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird

Aliquot sequence-related	Untouchable Amicable Sociable Betrothed

Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant

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centered	Centered pentachoric Squared triangular

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By properties of σ(n)	Abundant Almost perfect Arithmetic Colossally abundant Descartes Hemiperfect Highly abundant Highly composite Hyperperfect Multiply perfect Perfect Practical number Primitive abundant Quasiperfect Refactorable Sublime Superabundant Superior highly composite Superperfect

By properties of Ω(n)	Almost prime Semiprime

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