highly abundant number
| Divisibility-based sets of integers |
| Form of factorization: |
| Prime number |
| Composite number |
| Powerful number |
| Square-free number |
| Achilles number |
| Constrained divisor sums: |
| Perfect number |
| Almost perfect number |
| Quasiperfect number |
| Multiply perfect number |
| Hyperperfect number |
| Unitary perfect number |
| Semiperfect number |
| Primitive semiperfect number |
| Practical number |
| Numbers with many divisors: |
| Abundant number |
| Highly abundant number |
| Superabundant number |
| Colossally abundant number |
| Highly composite number |
| Superior highly composite number |
| Other: |
| Deficient number |
| Weird number |
| Amicable number |
| Friendly number |
| Sociable number |
| Solitary number |
| Sublime number |
| Harmonic divisor number |
| Frugal number |
| Equidigital number |
| Extravagant number |
| See also: |
| Divisor function |
| Divisor |
| Prime factor |
| Factorization |
In mathematics, a highly abundant number is a natural number where the sum of its divisors (including itself) is greater than the sum of the divisors of any natural number less than it.
Highly abundant numbers and several similar classes of numbers were first introduced by Pillai (1943), and early work on the subject was done by Alaoglu and Erdős (1944). Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any N is at least proportional to log2 N. They also proved that 7200 is the largest powerful highly abundant number, and therefore the largest highly abundant number with odd sum of divisors.
Formal definition and examples
Formally, a natural number n is called highly abundant if and only if for all natural numbers m < n,
- σ(n) > σ(m)
where σ denotes the sum-of-divisors function. The first few highly abundant numbers are
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.
Relations with other sets of numbers
Some sources report that all factorials are highly abundant numbers, but this is incorrect.
- 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 numbers are highly abundant, and asked whether there are infinitely many highly abundant numbers that are not superabundant. This question was answered affirmatively by Nicolas (1969).
Despite the terminology, not all highly abundant numbers are abundant numbers. In particular, none of the first seven highly abundant numbers are abundant.
References
- Alaoglu, L.; Erdős, P. (1944). "On highly composite and similar numbers.". Transactions of the American Mathematical Society 56: 448–469. MR0011087.
- Nicolas, Jean-Louis (1969). "Ordre maximal d'un élément du groupe Sn des permutations et "highly composite numbers"". Bull. Soc. Math. France 97: 129–191. MR0254130.
- Pillai, S. S. (1943). "Highly abundant numbers". Bull. Calcutta Math. Soc. 35: 141–156. MR0010560.
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