In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number itself. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example.
The first 28 abundant numbers are: For example, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, and 12, whose sum is 36. Because 36 is more than 24, the number 24 is abundant. Its abundance is 36 − 24 = 12.
Properties
The smallest odd abundant number is 945.
The smallest abundant number not divisible by 2 or by 3 is 5391411025 whose distinct prime factors are 5, 7, 11, 13, 17, 19, 23, and 29. 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
Infinitely many even and odd abundant numbers exist.
The set of abundant numbers has a non-zero natural density. 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 multiple of a perfect number is abundant. For example, every multiple of 6 is abundant because
Every multiple of an abundant number is abundant. For example, every multiple of 20 is abundant because
Every integer greater than 20161 can be written as the sum of two abundant numbers.
Numbers whose sum of proper factors equals the number itself are called perfect numbers, while numbers whose sum of proper factors is less than the number itself are called deficient numbers. The first known classification of numbers as deficient, perfect or abundant was by Nicomachus in his Introductio Arithmetica, which described abundant numbers as like deformed animals with too many limbs. The abundancy index of n is the ratioσ/n. Distinct numbers n1, n2,... with the same abundancy index are called friendly numbers. The sequence of least numbers n such that σ > kn, in which a2 = 12 corresponds to the first abundant number, grows very quickly. The smallest odd integer with abundancy index exceeding 3 is 1018976683725 = 33 × 52 × 72 × 11 × 13 × 17 × 19 × 23 × 29. If p = is a list of primes, then p is termed abundant if some integer composed only of primes in p is abundant. A necessary and sufficient condition for this is that the product of pi/ be at least 2.
