Define the harmonic mean of the divisors of 
where 
is the divisor function (the number of divisors of 
).
For 
, 2, ..., the values of 
are then 1, 4/3, 3/2, 12/7, 5/3, 2, 7/4, 32/15, 27/13, 20/9, ... (OEIS A099377 and A099378).
If 
is a perfect number, 
is an integer.
Ore conjectured that if 
is odd, then 
is not an integer. This implies that no odd perfect numbers exist.
See also
Harmonic Divisor Number,
Harmonic Mean,
Odd Perfect Number,
Perfect Number Explore with Wolfram|Alpha
References
Sloane, N. J. A. Sequences A099377 and A099378 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Ore's Conjecture Cite this as:
Weisstein, Eric W. "Ore's Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OresConjecture.html
Subject classifications
