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Odd abundant number facts for kids

Kids Encyclopedia Facts
Abundant number Cuisenaire rods 12
Demonstration, with Cuisenaire rods, of the abundance of the number 12

An odd abundant number is an odd number n 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

945+630n

gives 62 odd abundant numbers, but fails when

n=62.

The second formula

3465+2310n

gives 192 abundant numbers, but fails when

n=192

Furthermore, the last formula

2446903305+1631268870n

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 n. In other words, they are abundant numbers that do not divide any integer from 2 to n.

n first n-rough abundant number Factorization
2 12
3 945
5 5391411025
7 20169691...76756331
11 112 × 132 × 17 × 19 × 23 × ... × 103 × 107 × 109 × 113 × 127 × 131 × 137
13 132 × 172 × 19 × 23 × ... × 223 × 227
17 172 × 192 × 232 × 29 × 31 × ... × 347 × 349
19 192 × 232 × 292 × 31 × 37 × ... × 491 × 499

See also

Kids robot.svg In Spanish: Número abundante para niños

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