Highly abundant number Cuisenaire rods 8

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, a highly abundant number is a

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number. Highly abundant numbers and several similar classes of numbers were first introduced by , and early work on the subject was done by . Alaoglu and Erdős tabulated all highly abundant numbers up to 10⁴, and showed that the number of highly abundant numbers less than any is at least proportional to .

Formal definition and examples

Formally, a natural number ''n'' is called highly abundant

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...

for all natural numbers ''m'' < ''n'', :

\sigma(n) > \sigma(m)

where σ denotes the sum-of-divisors function. The first few highly abundant numbers are : 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, ... . For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant because σ(8) = 8 + 4 + 2 + 1 = 15 is larger than all previous values of σ. The only odd highly abundant numbers are 1 and 3.

Relations with other sets of numbers

Although the first eight

factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times ...

s are highly abundant, not all factorials are highly abundant. For example, :σ(9!) = σ(362880) = 1481040, but there is a smaller number with larger sum of divisors, :σ(360360) = 1572480, so 9! is not highly abundant. Alaoglu and Erdős noted that all

superabundant number In mathematics, a superabundant number is a certain kind of natural number. A natural number is called ''superabundant'' precisely when, for all : :\frac < \frac where denotes the abundant number In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total ...

s. In particular, none of the first seven highly abundant numbers (1, 2, 3, 4, 6, 8, and 10) is abundant. Along with 16, the ninth highly abundant number, these are the only highly abundant numbers that are not abundant. 7200 is the largest powerful number that is also highly abundant: all larger highly abundant numbers have a prime factor that divides them only once. Therefore, 7200 is also the largest highly abundant number with an odd sum of divisors., pp. 464–466.

Notes

References

* * * {{Classes of natural numbers Divisor function Integer sequences