Odd abundant number

2017-04-25T01:09:02Z

67.40.215.54: I corrected the reference from 'the next 12 numbers' to 'the next 11 numbers'

An '''odd abundant number''' is an [[odd number]] <math>n</math> that its [[sum-of-divisors|sum-of divisors]] greater than the [[double|twice]] of itself.
==Examples==
*The first example is 945 (''[[3 (number)|3]]<sup>3</sup>× [[5 (number)|5]]× [[7 (number)|7]]''). Its [[prime number|prime]] [[factor|factors]] are [[3 (number)|3]], [[5 (number)|5]], and [[7 (number)|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 [http://oeis.org/A005231/b005231.txt, A005231].

==Formulas==
The following [[formula]]

<math>945+630n</math><ref>{{Cite web|url=http://ms.appliedprobability.org/data/files/Articles%2038/38-1-2.pdf|title=More Odd Abundant Sequences|first=|last=|date=2005|website=|publisher=JAY. SCHIFFMAN|accessdate=2017-01-2}}</ref>
presents 62 [[abundant number]]s, but it fails if

<math>n\le62</math>.

The second formula

<math>3465+2310n</math><ref>{{Cite web|url=http://ms.appliedprobability.org/data/files/Articles%2038/38-1-2.pdf|title=More Odd Abundant Sequences|first=|last=|date=2005|website=|publisher=JAY. SCHIFFMAN|accessdate=2017-01-26}}</ref>
presents 192 [[abundant number]]s, but fails if

<math>n\le192</math>

The [[third]] formula

<math>2446903305+1631268870n</math> <ref>{{Cite web|url=http://ms.appliedprobability.org/data/files/Articles%2038/38-1-2.pdf|title=More Odd Abundant Sequences|first=|last=|date=2005|website=|publisher=JAY. SCHIFFMAN|accessdate=2017-01-26}}</ref>

fails if <math>n\le135939073</math>.

==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 10<sup>30<sup>.

==References==

{{Reflist}}

{{Math-stub}}

[[Category:Integer sequences]]

TCS Wiki - User contributions [en]

Odd abundant number