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 OEIS).

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:

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

5 is an integer, making 140 a harmonic divisor number.

Read more about Harmonic Divisor Number: Harmonic Divisor Numbers and Perfect Numbers, Bounds and Computer Searches

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