Ore's harmonic number

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

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.

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 $i$ th 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

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.

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