Highly abundant number

Highly abundant numbers are a type of natural number. Any natural number n is called highly abundant when a certain equation is true.

[math]\displaystyle{ \sigma(n) \gt \sigma(m) }[/math]

In this equation, m is every integer less than n. σ is the sum of every positive divisor of that number.

Example

An example would be to use the number 5. σ of 5 is 5 + 1 = 6. Every σ less than 5, however, is 4 + 2 + 1 = 7. 7 is greater than 6. This makes 5 not a highly abundant number.

History

Highly abundant numbers were first learned about by Subbayya Sivasankaranarayana Pillai in 1943. Work on the subject was done by Paul Erdős and Leonidas Alaoglu in 1944. Alaoglu and Erdős discovered every highly abundant number up to 10⁴. They also showed that the number of highly abundant numbers less than N is proportional to log² N.

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).

There are only two odd highly abundant numbers. They are 1 and 3.

Similarities with other numbers

The first eight factorials are highly abundant. However, 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.

This makes 9! is not highly abundant.

Alaoglu and Erdős discovered that all superabundant numbers are also highly abundant. There are, however, an infinite number of highly abundant numbers that are not superabundant numbers. This was proven by Jean-Louis Nicolas in 1969.

7200 is the largest powerful number that is also highly abundant. This is because every highly abundant number that is larger has a prime factor that divides them only once.

Highly Abundant Number Media

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  Sums of the divisors, in Cuisenaire rods, of the first six highly abundant numbers

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 S_n des permutations et "highly composite numbers"". Bull. Soc. Math. France. 97: 129–191. doi:10.24033/bsmf.1676. MR 0254130.
• Pillai, S. S. (1943). "Highly abundant numbers". Bull. Calcutta Math. Soc. 35: 141–156. MR 0010560.