Harmonic divisor number

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A harmonic divisor number, or Ore number, is a number 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

Four of these listed are also perfect numbers, and like perfect numbers, harmonic divisor numbers tend to be 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, 6 is a harmonic divisor number because its divisors,

1, 2, 3, 6

have a harmonic mean of

[\left (\frac+\frac+\frac+\frac}\right)=2]

Harmonic divisor number

See also