Ore's harmonic number

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The term harmonic number has multiple meanings. For other meanings, see harmonic number (disambiguation).

In mathematics, Ore's harmonic numbers, defined by Øystein Ore in 1948, are defined as the sequence of positive integers such that, for each number M in the sequence, the harmonic mean of the positive divisors of M is itself an integer. The first few Ore harmonic numbers are 1, 6, 28, 140, 270, 496, 672, 1638, … (sequence A001599 in OEIS).

As an example, consider the number 140 which has the positive divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. The harmonic mean of these divisors is

$\frac{12}{1+\frac{1}{2}+\frac{1}{4}+\frac{1}{5}+\frac{1}{7}+\frac{1}{10} +\frac{1}{14}+\frac{1}{20}+\frac{1}{28}+\frac{1}{35}+\frac{1}{70}+\frac{1}{140}}$

which equals 5, an integer, making 140 an Ore harmonic number.

[edit] Harmonic 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; see Bogomolny for a proof. 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 an Ore harmonic number.

Ore conjectured that no odd harmonic numbers exist other than 1. If the conjecture is true, this would imply the nonexistence of odd perfect numbers.

[edit] Bounds and computer searches

W. H. Mills (unpublished; see Muskat) showed that any odd harmonic number must have a prime power factor greater than 10⁷, and Cohen showed that any such number must have at least three different prime factors.

Cohen, Goto, and others starting with Ore himself have performed computer searches listing all small Ore harmonic numbers. From these results, lists are known of all Ore harmonic numbers up to 2×10⁹, and all Ore harmonic numbers for which the harmonic mean of the divisors is at most 300.