A harmonic divisor number, or Ore number, is a number whose divisors, averaged in a harmonic mean, results in an integer. The first few harmonic divisor numbers are

1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190

Three of these listed are also perfect numbers, and like perfect numbers, harmonic divisor numbers tend to even numbers, at least in the range observed. In 1972, W.H. Mills proved that, besides 1, there are no odd harmonic divisor numbers with prime power factors less than 10⁷.

For example, 496 is a harmonic divisor number because 10, its number of divisors, divided by the sum of the reciprocals of its divisors, 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496, (the harmonic mean), yields an integer, 5 in this case.

See also

• primitive semiperfect number