Harmonic divisor number

In mathematics, a harmonic divisor number or Ore number is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are

1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 Template:OEIS.

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 four divisors: 1, 2, 3, and 6. Their harmonic mean is an integer: $$\frac{4}{\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{6}}=2.$$ 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 $$\frac{12}{\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{5}+\frac{1}{7}+\frac{1}{10}+\frac{1}{14}+\frac{1}{20}+\frac{1}{28}+\frac{1}{35}+\frac{1}{70}+\frac{1}{140}}=5.$$ Since 5 is an integer, 140 is a harmonic divisor number.

Factorization of the harmonic mean

The harmonic mean Template:Math of the divisors of any number Template:Mvar can be expressed as the formula $$H(n)=\frac{n{\sigma }_0(n)}{{\sigma }_1(n)}$$ where Template:Math is the [[divisor function|sum of Template:Mvarth powers of the divisors]] of Template:Mvar: Template:Math is the number of divisors, and Template:Math is the sum of divisors Template:Harv. All of the terms in this formula are multiplicative, so that the harmonic mean Template:Math is also multiplicative. It follows that, for any positive integer Template:Mvar, the harmonic mean Template:Math can be expressed as the product of the harmonic means of the prime powers in the factorization of Template:Mvar.

For instance, we have $$H(4)=\frac{3}{1+\frac{1}{2}+\frac{1}{4}}=\frac{12}{7},$$ $$H(5)=\frac{2}{1+\frac{1}{5}}=\frac{5}{3},$$ $$H(7)=\frac{2}{1+\frac{1}{7}}=\frac{7}{4},$$ and $$H(140)=H(4\cdot 5\cdot 7)=H(4)\cdot H(5)\cdot H(7)=\frac{12}{7}\cdot \frac{5}{3}\cdot \frac{7}{4}=5.$$

Harmonic divisor numbers and perfect numbers

Demonstration, with Cuisenaire rods, of the perfection of the number 6

For any integer Template:Mvar, as Ore observed, the product of the harmonic mean and arithmetic mean of its divisors equals Template:Mvar itself, as can be seen from the definitions. Therefore, Template:Mvar is harmonic, with harmonic mean of divisors Template:Mvar, if and only if the average of its divisors is the product of Template:Mvar with a unit fraction Template:Math.

Ore showed that every perfect number is harmonic. To see this, observe that the sum of the divisors of a perfect number Template:Mvar is exactly Template:Math; therefore, the average of the divisors is Template:Math, where Template:Math denotes the number of divisors of M. For any Template:Mvar, Template:Math is odd if and only if Template:Mvar is a square number, for otherwise each divisor Template:Mvar of Template:Mvar can be paired with a different divisor Template:Math. 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 Template:Math where Template:Math. Therefore, for a perfect number Template:Mvar, Template:Math is even and the average of the divisors is the product of Template:Mvar with the unit fraction Template:Math; thus, Template:Mvar 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

W. H. Mills (unpublished; see Muskat) showed that any odd harmonic divisor number above 1 must have a prime power factor greater than 10⁷, and Cohen showed that any such number must have at least three different prime factors. Template:Harvtxt showed that there are no odd harmonic divisor numbers smaller than 10²⁴.

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 × 10⁹, and all harmonic divisor numbers for which the harmonic mean of the divisors is at most 300.

References

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