Abundant number

In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. 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. A number n for which the sum of divisors σ(n) > 2n, or, equivalently, the sum of proper divisors (or aliquot sum) s(n) > n. Abundance is the value σ(n) − 2n (or s(n) − n). The first 28 abundant numbers are: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, ... . For example, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, and 12, whose sum is 36. Because 36 is greater than 24, the number 24 is abundant. Its abundance is 36 − 24 = 12. 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 for sufficiently large k. Every multiple of a perfect number (except the perfect number itself) is abundant. For example, every multiple of 6 greater than 6 is abundant because Every multiple of an abundant number is abundant. For example, every multiple of 20 (including 20 itself) is abundant because Consequently, infinitely many even and odd abundant numbers exist. Furthermore, 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. An abundant number which is not the multiple of an abundant number or perfect number (i.e.