What is an Abundant Number?

An Abundant Number is a positive integer for which the sum of all its Proper divisors (factors) is more than the original number.

Abundant number is also called Excessive Numbers.

Examples of Abundant Numbers

Let us understand Abundant Number with a few examples.

Example 1. 20 is an Abundant number.

Proper divisors of 20 = 1, 2, 4, 5, and 10.

Sum = 1 + 2 + 4 + 5 + 10 = 22

As 22 > 20,

Therefore, 20 is an Abundant number or an Excessive number.

Example 2. 114 is an Abundant number.

Proper divisors of 114 = 1, 2, 3, 6, 19, 38, and 57.

Sum – 1 + 2 + 3 + 6 + 19 + 38 + 57 = 126

As 126 > 114,

Therefore, 114 is an Abundant number.

There are infinitely many Abundant numbers.

1. No Prime Number is an Abundant number.

2. All the multiples of Abundant numbers are also Abundant numbers.

3. Every proper multiple of a perfect number is an Abundant number.

4. Every integer greater than 20161 can be written as the sum of two abundant numbers.

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,….

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