Highly abundant number

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",

:$sigma(n)\; >\; sigma(m)$

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, ... OEIS|id=A002093.

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

*cite journal
author = Alaoglu, L.; Erdős, P.
title = On highly composite and similar numbers
journal = Transactions of the American Mathematical Society
volume = 56
year = 1944
pages = 448–469
id = MathSciNet | id = 0011087
doi = 10.2307/1990319

*cite journal
author = Nicolas, Jean-Louis
title = Ordre maximal d'un élément du groupe "S_n" des permutations et "highly composite numbers"
journal = Bull. Soc. Math. France
volume = 97
year = 1969
pages = 129–191
id = MathSciNet | id = 0254130
url = http://www.numdam.org/item?id=BSMF_1969__97__129_0

*cite journal
author = Pillai, S. S.
authorlink = Subbayya Sivasankaranarayana Pillai
title = Highly abundant numbers
journal = Bull. Calcutta Math. Soc.
volume = 35
year = 1943
pages = 141–156
id = MathSciNet | id = 0010560