# Harmonic divisor 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:

1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 (sequence A001599 in the OEIS).

## Examples

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

${\frac {4}{{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{6}}}}=2.$ The number 140 has divisors 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Their harmonic mean is:

${\frac {12}{{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{5}}+{\frac {1}{7}}+{\frac {1}{10}}+{\frac {1}{14}}+{\frac {1}{20}}+{\frac {1}{28}}+{\frac {1}{35}}+{\frac {1}{70}}+{\frac {1}{140}}}}=5$ 5 is an integer, making 140 a harmonic divisor number.

## Factorization of the harmonic mean

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

$H(n)={\frac {n\sigma _{0}(n)}{\sigma _{1}(n)}}$ where σi(n) is the sum of ith powers of the divisors of n: σ0 is the number of divisors, and σ1 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 of the prime powers in the factorization of n.

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

$H(140)=H(4\cdot 5\cdot 7)=H(4)\cdot H(5)\cdot H(7)={\frac {12}{7}}\cdot {\frac {5}{3}}\cdot {\frac {7}{4}}=5.$ 