TOPICS
Ore's Conjecture
Ore's conjecture asserts that the harmonic mean 
of the positive divisors
of an odd number 
is not an integer (Ore 1948).
For a positive integer 
, this harmonic mean is
 |
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.
The restriction 
excludes 
.
Ore's conjecture implies that no odd perfect numbers
exist.
See also
Harmonic Divisor Number, Harmonic Mean, Odd Perfect Number, Perfect NumberExplore with Wolfram|Alpha
References
Ore, Ø. "On the Averages of the Divisors of a Number." Amer. Math. Monthly 55, 615-619, 1948. https://doi.org/10.2307/2305616.Sloane, N. J. A. Sequences A099377 and A099378 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Ore's ConjectureCite this as:
Weisstein, Eric W. "Ore's Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OresConjecture.html