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Encyclopedia > Superabundant number

In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. Formally, a natural number n is called superabundant iff for any m < n, Mathematics is the study of quantity, structure, space and change. ... Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is... ↔ ⇔ ≡ For other possible meanings of iff, see IFF. In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. Although P iff Q is most standard, common alternative phrases include Q is necessary and sufficient for P and P...

where σ denotes the divisor 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 OEIS); superabundant numbers are closely related to highly composite numbers. In mathematics the divisor function σa(n) is defined as the sum of the ath powers of the divisors of n, or The notations d(n) and (the tau function) are also used to denote σ0(n), or the number of divisors of n. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ... A highly composite number is a positive integer which has more divisors than any positive integer below it. ...


Superabundant numbers were first defined in [AlaErd44].

Contents


Properties

Leonidas Alaoglu and Paul Erdős proved [AlaErd44] that if n is superabundant, then there exist a2, ..., ap such that Paul ErdÅ‘s Paul ErdÅ‘s (March 26, 1913 – September 20, 1996) was an immensely prolific and famously eccentric mathematician who, with hundreds of collaborators, worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory and probability theory. ...

and

In fact, ap is nearly always 1.


It can also be shown that all superabundant numbers are Harshad numbers. A Harshad number, or Niven number, is an integer that is divisible by the sum of its digits in a given number base. ...


Also see

In mathematics, a highly abundant number is a certain kind of natural number. ... In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. ... In mathematics, a deficient number or defective number is a number n for which σ(n) < 2n. ...

External links

  • MathWorld: Superabundant number

References

  • [AlaErd44] - Leonidas Alaoglu and Paul Erdős, On Highly Composite and Similar Numbers, Trans. AMS 56, 448-469 (1944)

  Results from FactBites:
 
Pythagorean Views on Numbers (1620 words)
Perfect are such as 6 and 28; equal to the number of their parts; as 28- half is 14, a fourth is 7, a seventh is 4, a fourteenth part is 2, and the twenty-eighth is 1, which quotients added together are 28.
In Superabundant, as 12, the whole surpasses the aggregate of its parts; thus the sixth is 2, a fourth is 3, a third is 4, a half is 6, and a twelfth is 1; and the aggregate is 16, or more than 12.
The deficient numbers resembled Cyclops, who had but one eye; whilst the perfect numbers have the temperament of a middle limit, and are the emulators of Virtue, a medium between excess and defect, not the summit, as some ancients falsely thought.
Pearl and Number Symbolism (1648 words)
Pagan interpretations of number symbols usually are said to arise with Pythagoras, a Greek philosopher who sought wisdom in math and geometry, hence the theorem for calculating the triangle's longest side which you probably memorized.
When examining numbers themselves to explain this relationship, Pythagoras discovered "divine tetraktus," the fact that the number 10 is the sum of the first four numbers and those numbers corresponded to the geometric forms necessary to construct physical space (1=point, 2=line, 3=planar surfaces, 4=geometric solids).
Some numbers' factors added up to the numbers themselves (e.g., 1, 2, and 3 are the factors of 6), and such numbers were considered "perfect" because they were exactly harmonious with their divisors.
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