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
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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 Web Resource. https://mathworld.wolfram.com/OresConjecture.html
Subject classifications
