Odd abundant number facts for kids

Demonstration, with Cuisenaire rods, of the abundance of the number 12

An odd abundant number is an odd number letting its sum-of divisors greater than the twice of itself.

Examples

• The first example is 945 (3³× 5× 7) and having the abundance of 30. And the next twelve are

1575, 2205, 2835, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615.

• Odd abundant numbers below 500000 are in On-Line Encyclopedia of Integer Sequences A005231.

Formula

The formula

gives 62 odd abundant numbers, but fails when

.

The second formula

gives 192 abundant numbers, but fails when

Furthermore, the last formula

are abundant numbers, until , it appears to be the first counterexample.

Special Properties

• An abundant twin is a pair of (n, n+1) and they are both abundant numbers. Abundant twins are usually less common. The first three pairs of abundant twins are (5984, 5985) (11024,11025) (81080, 81081).
• An calculation was given by Iannucci shows how to find the smallest abundant number not divisible by the first n primes.
• An abundant number with abundance 1 is called a quasiperfect number, although none have yet been found. A quasiperfect number must be an odd square number having a value above 10³⁰

^n-rough

^{A n-rough abundant number is an abundant number not divisible by primes under . In other words, they are abundant numbers that do not divide any integer from 2 to n.}

^{See also}

^{In Spanish: Número abundante para niños}