is called harmonic divisor number
harmonic mean of its divisors is an integer.
This is equivalent to say that the average of the divisors of
is an integer.
Harmonic divisor numbers are also called harmonic numbers, for brevity, or
Ore numbers, after O.Ore who studied them.
O.Ore proved that all the perfect numbers are also harmonic and
conjectured that 1 is the only odd harmonic number. This conjecture
has been verified by G.L.Cohen et al. for and if true,
it will imply that no odd perfect numbers exist.
Jaycob Coleman has observed that all the Ore numbers up to are
also practical numbers and conjectured this holds
T. Goto and K. Okeya have computed a list of the 937 harmonic numbers
up to .
The first harmonic numbers are
1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 18620, 27846, 30240, 32760, 55860, 105664, 117800, 167400, 173600, 237510 more terms
Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
A graph displaying how many harmonic numbers are multiples of the primes p
from 2 to 71. In black the ideal line 1/p