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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

Sociable number

Sublime number

Harmonic divisor 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.

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

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.

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).

• 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 S_n 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.