Definition[change | change source]
Any natural number n is called superabundant when a certain equation is true.
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
Properties[change | change source]
where pi is the i-th prime number, and
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 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
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[change | change source]
- 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.