# Odd abundant number

An odd abundant number is an odd number ${\displaystyle n}$ that its sum-of divisors greater than the twice of itself.

## Examples

• The first example is 945 (33× 5× 7). Its prime factors are 3, 5, and 7. The next following eleven odd abundant numbers 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.

## Formulas

The following formula

${\displaystyle 945+630n}$[1] presents 62 abundant numbers, but it fails if

${\displaystyle n\leq 62}$.

The second formula

${\displaystyle 3465+2310n}$[2] presents 192 abundant numbers, but fails if

${\displaystyle n\leq 192}$

The third formula

${\displaystyle 2446903305+1631268870n}$ [3]

fails if ${\displaystyle n\leq 135939073}$.

## Properties

• 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.

## References

1. "More Odd Abundant Sequences" (PDF). JAY. SCHIFFMAN. 2005. Retrieved 2017-01-2. Check date values in: |accessdate= (help)
2. "More Odd Abundant Sequences" (PDF). JAY. SCHIFFMAN. 2005. Retrieved 2017-01-26.
3. "More Odd Abundant Sequences" (PDF). JAY. SCHIFFMAN. 2005. Retrieved 2017-01-26.