Superabundant number

A superabundant number is a type of natural number. They were defined by Leonidas Alaoglu and Paul Erdős in 1944.

Definition

Any natural number n is called superabundant when a certain equation is true.

${\displaystyle {\frac {\sigma (m)}{m}}<{\frac {\sigma (n)}{n}}}$

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

An example would be to use the number 9. For every number less than 8, the sigma is 1, 3, 4, 7, 6, 10, 8 and 16. (σ(m))/m is 16/9. (σ(n))/n is equal to 13/9. 13/9 is less than 16/9. This makes 9 not a superabundant number

The first few superabundant numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, ... (sequence A004394 in the OEIS).

Properties

Leonidas Alaoglu and Paul Erdős proved that if n is superabundant, then there is a k and a1, a2, ..., ak such that

${\displaystyle n=\prod _{i=1}^{k}(p_{i})^{a_{i}}}$

where pi is the i-th prime number, and

${\displaystyle a_{1}\geq a_{2}\geq \dotsb \geq a_{k}\geq 1.}$

Basically, they proved that if a number is superabundant, the exponent of a larger prime number is never bigger than a smaller prime number during prime decomposition(the process of a composite number become smaller prime numbers). All primes from 0 to ${\displaystyle p_{k}}$ are also factors of n. The equation says that a superabundant number has to be an even integer. It also is a multiple of the k-th primorial ${\displaystyle p_{k}\#.}$

Superabundant numbers are like highly composite numbers. Not all superabundant numbers are highly composite numbers, though.

Alaoglu and Erdős observed that all superabundant numbers are also highly abundant.

References

• Briggs, Keith (2006), "Abundant numbers and the Riemann hypothesis", Experimental Mathematics, 15: 251–256.
• Akbary, Amir; Friggstad, Zachary (2009), "Superabundant numbers and the Riemann hypothesis", American Mathematical Monthly, 116 (3): 273–275, doi:10.4169/193009709X470128.
• Alaoglu, Leonidas; Erdős, Paul (1944), "On highly composite and similar numbers", Transactions of the American Mathematical Society, American Mathematical Society, 56 (3): 448–469, doi:10.2307/1990319, JSTOR 1990319.