Editing Harmonic divisor number

{{Short description|Positive integer whose divisors have a harmonic mean that is an integer}}
{{About|harmonic divisor numbers|meanings of harmonic number|harmonic number (disambiguation)}}

In [[mathematics]], a '''harmonic divisor number''' or '''Ore number''' is a positive [[integer]] whose [[divisor]]s 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}}.
Harmonic divisor numbers were introduced by [[Øystein Ore]], who showed that every [[perfect number]] is a harmonic divisor number and conjectured that there are no odd harmonic divisor numbers other than 1.

==Examples==
The number 6 has the four divisors 1, 2, 3, and 6. Their harmonic mean is an integer:
<math display="block"> \frac{4}{\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{6}}=2.</math>
Thus 6 is a harmonic divisor number.  Similarly, the number 140 has divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Their harmonic mean is
<math display="block">
 \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>
Since 5 is an integer, 140 is a harmonic divisor number.

==Factorization of the harmonic mean==
The harmonic mean {{math|''H''(''n'')}} of the divisors of any number {{mvar|n}} can be expressed as the formula
<math display="block">H(n) = \frac{n\sigma_0(n)}{\sigma_1(n)}</math>
where {{math|''&sigma;''<sub>''i''&thinsp;</sub>(''n'')}} is the [[divisor function|sum of {{mvar|i}}th powers of the divisors]] of {{mvar|n}}: {{math|''&sigma;''<sub>0</sub>}} is the number of divisors, and {{math|''&sigma;''<sub>1</sub>}} is the sum of divisors {{harv|Cohen|1997}}.
All of the terms in this formula are [[multiplicative function|multiplicative]] but not [[Completely multiplicative function|completely multiplicative]].
Therefore, the harmonic mean {{math|''H''(''n'')}} is also multiplicative.
This means that, for any positive integer {{mvar|n}}, the harmonic mean {{math|''H''(''n'')}} can be expressed as the product of the harmonic means of the [[prime power]]s in the [[Integer factorization|factorization]] of {{mvar|n}}.

For instance, we have
<math display="block">H(4) = \frac{3}{1+\frac{1}{2}+\frac{1}{4}}=\frac{12}7,</math>
<math display="block">H(5) = \frac{2}{1+\frac{1}{5}} = \frac53,</math>
<math display="block">H(7) = \frac{2}{1+\frac{1}{7}} = \frac74,</math>
and
<math display="block">H(140) = H(4 \cdot 5 \cdot 7) = H(4)\cdot H(5)\cdot H(7) = \frac{12}{7}\cdot \frac{5}{3}\cdot \frac{7}{4} = 5.</math>

== Harmonic divisor numbers and perfect numbers ==
[[File:Perfect number Cuisenaire rods 6 exact.svg|thumb|Demonstration, with [[Cuisenaire rods]], of the perfection of the number 6]]

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 [[parity (mathematics)|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 [[Perfect_number#Even_perfect_numbers|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 ([[modular arithmetic|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 [[conjecture]]d that no odd harmonic divisor numbers exist other than 1. If the conjecture is true, this would imply the nonexistence of [[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 number|prime]] factors. {{harvtxt|Cohen|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&nbsp;×&thinsp;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
  | access-date = 2006-09-10| author-link = Alexander Bogomolny
 }}
*{{cite journal
  | last = Cohen | first = Graeme L.
  | url =https://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
  | doi-access = free
  }}
*{{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
  | doi-access=free
  | hdl=10453/13014
  | hdl-access=free
  }}
*{{cite web
  | author = Goto, Takeshi
  | url = http://www.ma.noda.tus.ac.jp/u/tg/harmonic-e.html
  | title = (Ore's) Harmonic Numbers
  | access-date = 2006-09-10
  | archive-date = 2006-01-06
  | archive-url = https://web.archive.org/web/20060106230537/http://www.ma.noda.tus.ac.jp/u/tg/harmonic-e.html
  | url-status = dead
  }}
* {{cite book |last=Guy | first=Richard K. | author-link=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
  | jstor = 2004277| doi-access = free
  }}
*{{cite journal
  | author = Ore, Øystein
  | author-link = 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
  | jstor = 2305616}}
* {{MathWorld |title=Harmonic Divisor Number |urlname=HarmonicDivisorNumber}}

{{Divisor classes}}
{{Classes of natural numbers}}

[[Category:Divisor function]]
[[Category:Integer sequences]]
[[Category:Number theory]]
[[Category:Perfect numbers]]