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.

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 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.

The only odd highly abundant numbers are 1 and 3.[1]
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 Jean-Louis 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.

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.[2]
Notes

See Alaoglu & Erdős (1944), p. 466. Alaoglu and Erdős claim more strongly that all highly abundant numbers greater than 210 are divisible by 4, but this is not true: 630 is highly abundant, and is not divisible by 4. (In fact, 630 is the only counterexample; all larger highly abundant numbers are divisible by 12.)

Alaoglu & Erdős (1944), pp. 464–466.

References

Alaoglu, L.; Erdős, P. (1944). "On highly composite and similar numbers" (PDF). Transactions of the American Mathematical Society 56 (3): 448–469. doi:10.2307/1990319. JSTOR 1990319. MR 0011087.
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. MR 0254130.
Pillai, S. S. (1943). "Highly abundant numbers". Bull. Calcutta Math. Soc. 35: 141–156. MR 0010560.

Mathematics Encyclopedia