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{{About|harmonic divisor numbers|meanings of harmonic number|harmonic number (disambiguation)}}
In [[mathematics]], a '''harmonic divisor number''', or '''Ore number''' (named after [[Øystein Ore]] who defined it in 1948), is a positive integer whose divisors have a [[harmonic mean]] that is an [[integer]]. The first few harmonic divisor numbers are


:[[1 (number)|1]], [[6 (number)|6]], [[28 (number)|28]], [[140 (number)|140]], [[270 (number)|270]], [[496 (number)|496]], 672, 1638, 2970, 6200, [[8128 (number)|8128]], 8190 {{OEIS|id=A001599}}.


For example, the harmonic divisor number 6 has the four divisors 1, 2, 3, and 6. Their harmonic mean is an integer:
Looking at playing a new visual game, read the cheat book. Most movie have a book a person can purchase separately. You may want in consider doing this but [http://Www.Tumblr.com/tagged/reading reading] it before the individual play, or even even although you are playing. If you are you looking for more about how to hack clash of clans; [http://circuspartypanama.com http://circuspartypanama.com], take a look at our own web-site. This way, you can possibly get the most out of your game be.<br><br>The underside line is, this is worth exploring if must make sure strategy games, especially when you're keen on Clash of Clans. Want recognize what opinions you possess, when you do.<br><br>To take pleasure from unlimited points, resources, silver coins or gems, you must download the clash of clans hack into tool by clicking in the button. Depending about the operating system that are generally using, you will need to run the downloaded content as administrator. Provide the log in ID and judge the device. Looking for this, you are ought to enter the number because of gems or coins that you prefer to get.<br><br>A very good method to please your kids with a gaming system and ensure they continue to exist fit is to obtain Wii. This console needs real task perform. Your children won't be positioned for hours on cure playing clash of clans hack. They really need to be moving around as a way to play the games for this particular system.<br><br>Second, when your husband decides to commit adultery, person creates a problem in which it forces you to cook some serious decisions. Step one turn within your Xbox sign through the dash board. It is unforgivable and as well , disappointing to say a minimum. I think we should start differentiating between currently the public interest, and some sort of proper definition of just what that means, and tips that the media pick out the public people 'd be interested in. Ford introduced the before anything else production woodie in 1929. The varieties in fingers you perform on the inside No-Limit Holdem vary as opposed to what all those in Restrain.<br><br>You will notice for yourself that this useful Money Compromise of Clans i fairly effective, completely invisible by the supervisor of the game, above all absolutely no price!<br><br>Seeking are playing a exhibiting activity, and you also don't possess knowledge of it, establish the problem stage to rookie. This should help you pick-up in the outstanding options that come when using the game and discover closer round the field. Should you set the software more than that, you are likely to get frustrated and never possess fun.
 
:<math> \frac{4}{\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{6}}=2.</math>
 
The number 140 has divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Their harmonic mean is:
 
:<math>
\frac{12}{\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{5}+\frac{1}{7}+\frac{1}{10}
+\frac{1}{14}+\frac{1}{20}+\frac{1}{28}+\frac{1}{35}+\frac{1}{70}+\frac{1}{140}}=5
</math>
 
5 is an integer, making 140 a harmonic divisor number.
 
== Harmonic divisor numbers and perfect numbers ==
 
For any integer ''M'', as Ore observed, the product of the harmonic mean and [[arithmetic mean]] of its divisors equals ''M'' itself, as can be seen from the definitions. Therefore, ''M'' is harmonic, with harmonic mean of divisors ''k'', if and only if the average of its divisors is the product of ''M'' with a [[unit fraction]] 1/''k''.
 
Ore showed that every [[perfect number]] is harmonic. To see this, observe that the sum of the divisors of a perfect number ''M'' is exactly ''2M''; therefore, the average of the divisors is ''M''(2/τ(''M'')), where τ(''M'') denotes the [[Divisor function|number of divisors]] of ''M''. For any ''M'', τ(''M'') is odd if and only if ''M'' is a [[square number]], for otherwise each divisor ''d'' of ''M'' can be paired with a different divisor ''M''/''d''. But, no perfect number can be a square: this follows from the known form of even perfect numbers and from the fact that odd perfect numbers (if they exist) must have a factor of the form ''q''<sup>α</sup> where α ≡ 1 (mod 4). Therefore, for a perfect number ''M'', τ(''M'') is even and the average of the divisors is the product of ''M'' with the unit fraction 2/τ(''M''); thus, ''M'' is a harmonic divisor number.
 
Ore conjectured that no odd harmonic divisor numbers exist other than 1. If the conjecture is true, this would imply the nonexistence of [[perfect number#Odd perfect numbers|odd perfect number]]s.
 
== Bounds and computer searches ==
 
[[W. H. Mills]] (unpublished; see Muskat) showed that any odd harmonic divisor number above 1 must have a prime power factor greater than 10<sup>7</sup>, and Cohen showed that any such number must have at least three different prime factors. Cohen and Sorli (2010) showed that there are no odd harmonic divisor numbers smaller than 10<sup>24</sup>.
 
Cohen, Goto, and others starting with Ore himself have performed computer searches listing all small harmonic divisor numbers. From these results, lists are known of all harmonic divisor numbers up to 2×10<sup>9</sup>, and all harmonic divisor numbers for which the harmonic mean of the divisors is at most 300.
 
== References ==
{{portal|Mathematics}}
 
*{{cite web
  | author = Bogomolny, Alexander
  | title = An Identity Concerning Averages of Divisors of a Given Integer
  | url = http://www.cut-the-knot.org/proofs/average.shtml
  | accessdate = 2006-09-10}}
 
*{{cite journal
  | author = Cohen, Graeme L.
  | url =http://www.ams.org/mcom/1997-66-218/S0025-5718-97-00819-3/S0025-5718-97-00819-3.pdf
  | title = Numbers Whose Positive Divisors Have Small Integral Harmonic Mean
  | journal = [[Mathematics of Computation]]
  | volume = 66
  | pages = 883–891
  | year = 1997
  | doi = 10.1090/S0025-5718-97-00819-3
  | issue = 218}}
 
*{{cite journal
  | last1=Cohen
  | first1=Graeme L.
  | last2=Sorli
  | first2=Ronald M.
  | year=2010
  | pages=2451
  | title=Odd harmonic numbers exceed 10<sup>24</sup>
  | volume=79
  | doi= 10.1090/S0025-5718-10-02337-9
  | journal=Mathematics of Computation
  | issn=0025-5718
  | issue=272}}  Posted electronically on April 9, 2010; to appear in print.
 
*{{cite web
  | author = Goto, Takeshi
  | url = http://www.ma.noda.tus.ac.jp/u/tg/harmonic-e.html
  | title = (Ore's) Harmonic Numbers
  | accessdate = 2006-09-10}}
 
* {{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=978-0-387-20860-2 | zbl=1058.11001 | at=B2 }}
 
*{{cite journal
  | title = On Divisors of Odd Perfect Numbers
  | author = Muskat, Joseph B.
  | journal = Mathematics of Computation
  | volume = 20
  | issue = 93
  | year = 1966
  | pages = 141–144
  | doi = 10.2307/2004277
  | publisher = American Mathematical Society
  | jstor = 2004277}}
 
*{{cite journal
  | author = Ore, Øystein
  | authorlink = Øystein Ore
  | title = On the averages of the divisors of a number
  | journal = [[American Mathematical Monthly]]
  | volume = 55
  | year = 1948
  | pages = 615–619
  | doi = 10.2307/2305616
  | issue = 10
  | publisher = Mathematical Association of America
  | jstor = 2305616}}
 
* {{MathWorld |title=Harmonic Divisor Number |urlname=HarmonicDivisorNumber}}
 
 
{{Divisor classes}}
{{Classes of natural numbers}}
 
[[Category:Divisor function]]
[[Category:Integer sequences]]
[[Category:Number theory]]

Latest revision as of 16:37, 26 October 2014


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