Abundant Numbers

Numbers whose sum of proper divisors exceeds the number itself

An abundant number is a natural number whose sum of proper divisors (all divisors except the number itself) is greater than the number. For example, 12 is abundant because its proper divisors are 1, 2, 3, 4, 6 and their sum (16) is greater than 12. The "abundance" of 12 is 16 − 12 = 4.

Classification of numbers by divisors

Based on the relationship between a number and the sum of its proper divisors, natural numbers are classified into three categories:

Deficient

Sum of divisors < n

Ejemplo: 8
Divisores: 1 + 2 + 4 = 7 < 8
Perfect

Sum of divisors = n

Ejemplo: 6
Divisores: 1 + 2 + 3 = 6
Abundant

Sum of divisors > n

Ejemplo: 12
Divisores: 1 + 2 + 3 + 4 + 6 = 16 > 12

Properties of abundant numbers

Abundant numbers possess several interesting properties that distinguish them in number theory:

First abundant The first abundant number is 12
Multiples Every multiple of an abundant number is also abundant
Sum of two abundants Every number greater than 20,161 can be expressed as the sum of two abundant numbers
Proportion Approximately 25% of natural numbers are abundant
Even abundants All even numbers greater than 46 are abundant (and many before that as well)
Smallest odd abundant The smallest odd abundant number is 945

The abundance of a number

The abundance of a number n is defined as A(n) = σ(n) − 2n, where σ(n) is the sum of all divisors of n (including n itself). If A(n) > 0, the number is abundant. It can also be calculated as the sum of proper divisors minus n.

Number Proper divisors Sum Abundance
12 1, 2, 3, 4, 6 16 +4
18 1, 2, 3, 6, 9 21 +3
20 1, 2, 4, 5, 10 22 +2
24 1, 2, 3, 4, 6, 8, 12 36 +12
30 1, 2, 3, 5, 6, 10, 15 42 +12
36 1, 2, 3, 4, 6, 9, 12, 18 55 +19
40 1, 2, 4, 5, 8, 10, 20 50 +10
48 1, 2, 3, 4, 6, 8, 12, 16, 24 76 +28
60 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 108 +48
70 1, 2, 5, 7, 10, 14, 35 74 +4

Superabundant numbers

A number n is superabundant if the ratio σ(n)/n is greater than σ(m)/m for all m < n, where σ(n) is the sum of all divisors of n. In other words, they are the numbers that "break the record" in the ratio between the sum of divisors and the number itself.

The first superabundant numbers are:

1 2 4 6 12 24 36 48 60 120 180 240 360 720 840 1,260

Notice that many superabundant numbers are highly composite (they have many divisors), such as 12, 24, 60, 120, 360, and 720. These numbers appear frequently in measurement systems (12 hours, 60 minutes, 360 degrees) precisely because of their large number of divisors.

The first 80 abundant numbers

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12 18 20 24 30 36 40 42 48 54 56 60 66 70 72 78 80 84 88 90 96 100 102 104 108 112 114 120 126 132 138 140 144 150 156 160 162 168 174 176 180 186 192 196 198 200 204 208 210 216 220 222 224 228 234 240 246 252 258 260 264 270 272 276 280 282 288 294 300 304 306 308 312 318 320 324 330 336 340 342

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