Abundant number

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In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. Here σ(n) is the divisor function: the sum of all positive divisors of n, including n itself. The value σ(n) − 2n is called the abundance of n. An equivalent definition is that the proper divisors of the number (the divisors except the number itself) sum to more than the number.

The first few abundant numbers (sequence A005101 in OEIS) are:

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, …

As an example, consider the number 24. Its divisors are 1, 2, 3, 4, 6, 8, 12 and 24, whose sum is 60. Because 60 is more than 2 × 24, the number 24 is abundant. Its abundance is 60 − 2 × 24 = 12.

The smallest odd abundant number is 945. Marc Deléglise showed in 1998 that the natural density of abundant numbers is between 0.2474 and 0.2480.

Infinitely many even and odd abundant numbers exist. Every proper multiple of a perfect number, and every multiple of an abundant number is abundant. Also, 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.

Closely related to abundant numbers are perfect numbers with σ(n) = 2n, and deficient numbers with σ(n) < 2n. The natural numbers were first classified as either deficient, perfect or abundant by Nicomachus in his Introductio Arithmetica (circa 100).