Properties
Leonidas Alaoglu and Paul Erdős (1944) proved that if n is superabundant, then there exist an i and a1, a2, ..., ai such that
where pl is the l-th prime number, and
Or to wit, if n is superabundant then the prime decomposition of n has decreasing exponents, the smaller the prime, the larger its exponent.
In fact, ai is equal to 1 except when n is 4 or 36.
Superabundant numbers are closely related to highly composite numbers. It is false that all superabundant numbers are highly composite numbers. In fact, only 449 superabundant and highly composite numbers are the same. For instance, 7560 is highly composite but not superabundant. Alaoglu and Erdős observed that all superabundant numbers are highly abundant. It is false that all superabundant numbers are Harshad numbers. The first exception is the 105th SA number, 149602080797769600. The digit sum is 81, but 81 does not divide evenly into this SA number.
Superabundant numbers are also of interest in connection with the Riemann hypothesis, and with Robin's theorem that the Riemann hypothesis is equivalent to the statement that
for all n greater than the largest known exception, the superabundant number 5040. If this inequality has a larger counterexample, proving the Riemann hypothesis to be false, the smallest such counterexample must be a superabundant number (Akbary & Friggstad 2009).
Read more about this topic: Superabundant Number
Famous quotes containing the word properties:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)