Odd abundant number facts for kids
An odd abundant number is an odd number letting its sum-of divisors greater than the twice of itself.
Contents
Examples
- 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 10^{30}
^{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.}
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} |
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Odd abundant number Facts for Kids. Kiddle Encyclopedia.