Abundant number
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. 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. Definition. A number "n" for which the sum of divisors "σ"("n") > 2"n", or, equivalently, the sum of proper divisors (or aliquot sum) "s"("n") > "n". Abundance is  the value "σ"("n") − "2n" (or "s"("n") − "n"). Examples. 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 greater than 24, the number 24 is abundant. Its abundance is 36 − 24 = 12. Related concepts. Numbers whose sum of proper factors equals the number itself (such as 6 and 28) 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" (circa 100 AD), which described abundant numbers as like deformed animals with too many limbs. The abundancy index of "n" is the ratio "σ"("n")/"n". Distinct numbers "n", "n", ... (whether abundant or not) with the same abundancy index are called friendly numbers. The sequence ("a") of least numbers "n" such that "σ"("n") > "kn", in which "a" = 12 corresponds to the first abundant number, grows very quickly . The smallest odd integer with abundancy index exceeding 3 is 1018976683725 = 3 × 5 × 7 × 11 × 13 ×  17 × 19 × 23 × 29. If p = ("p", ..., "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 "p"/("p" − 1) be > 2.