Properties
- The smallest odd abundant number is 945
- The smallest not divisible by 2 or by 3 is 5391411025 whose prime factors are 5, 7, 11, 13, 17, 19, 23, and 29 (sequence A047802 in OEIS). An algorithm given by Iannucci in 2005 shows how to find the smallest abundant number not divisible by the first k primes. If represents the smallest abundant number not divisible by the first k primes then for all we have:
- for k sufficiently large.
- Infinitely many even and odd abundant numbers exist. Marc Deléglise showed in 1998 that the natural density of the set of abundant numbers and perfect numbers is between 0.2474 and 0.2480.
- Every proper multiple of a perfect number, and every multiple of an abundant number, is abundant.
- Every integer greater than 20161 can be written as the sum of two abundant numbers.
- An abundant number which is not a semiperfect number is called a weird number; an abundant number with abundance 1 is called a quasiperfect number, however, none have yet been found.
Read more about this topic: Abundant 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)