# Harmonic divisor number

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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

## Contents

## Examples[edit]

For example, the harmonic divisor number 6 has the four divisors 1, 2, 3, and 6. Their harmonic mean is an integer:

The number 140 has divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Their harmonic mean is:

5 is an integer, making 140 a harmonic divisor number.

## Factorization of the harmonic mean[edit]

The harmonic mean *H*(*n*) of the divisors of any number n can be expressed as the formula

where *σ*_{i}(*n*) is the sum of ith powers of the divisors of n: *σ*_{0} is the number of divisors, and *σ*_{1} is the sum of divisors (Cohen 1997).
All of the terms in this formula are multiplicative, but not completely multiplicative.
Therefore, the harmonic mean *H*(*n*) is also multiplicative.
This means that, for any positive integer n, the harmonic mean *H*(*n*) can be expressed as the product of the harmonic means for the prime powers in the factorization of n.

For instance, we have

and

## Harmonic divisor numbers and perfect numbers[edit]

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 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*^{α} 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 odd perfect numbers.

## Bounds and computer searches[edit]

W. H. Mills (unpublished; see Muskat) showed that any odd harmonic divisor number above 1 must have a prime power factor greater than 10^{7}, and Cohen showed that any such number must have at least three different prime factors. Cohen & Sorli (2010) showed that there are no odd harmonic divisor numbers smaller than 10^{24}.

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^{9}, and all harmonic divisor numbers for which the harmonic mean of the divisors is at most 300.

## References[edit]

- Bogomolny, Alexander. "An Identity Concerning Averages of Divisors of a Given Integer". Retrieved 2006-09-10.
- Cohen, Graeme L. (1997). "Numbers Whose Positive Divisors Have Small Integral Harmonic Mean" (PDF).
*Mathematics of Computation*.**66**(218): 883–891. doi:10.1090/S0025-5718-97-00819-3. - Cohen, Graeme L.; Sorli, Ronald M. (2010). "Odd harmonic numbers exceed 10
^{24}".*Mathematics of Computation*.**79**(272): 2451. doi:10.1090/S0025-5718-10-02337-9. ISSN 0025-5718. - Goto, Takeshi. "(Ore's) Harmonic Numbers". Retrieved 2006-09-10.
- Guy, Richard K. (2004).
*Unsolved problems in number theory*(3rd ed.). Springer-Verlag. B2. ISBN 978-0-387-20860-2. Zbl 1058.11001. - Muskat, Joseph B. (1966). "On Divisors of Odd Perfect Numbers".
*Mathematics of Computation*.**20**(93): 141–144. doi:10.2307/2004277. JSTOR 2004277. - Ore, Øystein (1948). "On the averages of the divisors of a number".
*American Mathematical Monthly*.**55**(10): 615–619. doi:10.2307/2305616. JSTOR 2305616. - Weisstein, Eric W. "Harmonic Divisor Number".
*MathWorld*.