Odd abundant number facts for kids
An odd abundant number is a special kind of odd number. It's a number where if you add up all its divisors (numbers that divide it evenly, not including the number itself), the total is bigger than the number itself.
For example, let's look at the number 12. Its divisors are 1, 2, 3, 4, and 6. If you add them up (1+2+3+4+6), you get 16. Since 16 is greater than 12, 12 is an abundant number. However, 12 is an even number, so it's not an odd abundant number.
Contents
Examples of Odd Abundant Numbers
The first odd abundant number is 945.
- 945 can be broken down into its prime factors: 3 x 3 x 3 x 5 x 7.
- If you add up all its divisors (1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315, 945), the sum is 2016.
- Since 2016 is much larger than 945, 945 is an odd abundant number.
Here are the next twelve odd abundant numbers after 945:
- 1575
- 2205
- 2835
- 3465
- 4095
- 4725
- 5355
- 5775
- 5985
- 6435
- 6615
You can find a longer list of odd abundant numbers below 500,000 in the On-Line Encyclopedia of Integer Sequences A005231.
Special Properties of Abundant Numbers
Abundant Twins
An abundant twin is a pair of numbers that are right next to each other, and both of them are abundant numbers. These pairs are not very common.
The first three pairs of abundant twins are:
- (5984, 5985)
- (11024, 11025)
- (81080, 81081)
Quasiperfect Numbers
A quasiperfect number is a special type of abundant number. For these numbers, if you add up all their divisors (not including the number itself), the sum is exactly one more than the number itself.
For example, if a number was 100, and its divisors added up to 101, it would be a quasiperfect number.
- Even though mathematicians have looked for them, no quasiperfect numbers have ever been found.
- If a quasiperfect number exists, it must be an odd square number and be very, very large (bigger than 1030).
Images for kids
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This picture shows how the number 12 is abundant using Cuisenaire rods.
See also
In Spanish: Número abundante para niños
