Highly abundant number

In mathematics, a highly abundant number is a natural number 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 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, ... (sequence A002093 in the OEIS).

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

Although the first eight factorials 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 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 is abundant.

References

• Alaoglu, L.; Erdős, P. (1944). "On highly composite and similar numbers". Transactions of the American Mathematical Society. 56 (3): 448–469. doi:10.2307/1990319. JSTOR 1990319. MR 0011087.{{cite journal}}: CS1 maint: multiple names: authors list (link)

• Nicolas, Jean-Louis (1969). "Ordre maximal d'un élément du groupe S_n des permutations et "highly composite numbers"". Bull. Soc. Math. France. 97: 129–191. MR 0254130.

• Pillai, S. S. (1943). "Highly abundant numbers". Bull. Calcutta Math. Soc. 35: 141–156. MR 0010560.