Ore's conjecture asserts that the harmonic mean divisors
of an odd number integer (Ore 1948).
For a positive integer harmonic mean
is
where divisor function (the number of divisors
of
For A099377 and A099378 ).
If perfect number , integer .
The restriction odd
perfect numbers exist.
See also Harmonic Divisor Number ,
Harmonic Mean ,
Odd
Perfect Number ,
Perfect Number
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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 Conjecture
Cite this as:
Weisstein, Eric W. "Ore's Conjecture."
From MathWorld https://mathworld.wolfram.com/OresConjecture.html
Subject classifications
of the positive divisors
of an odd number 
is not an integer (Ore 1948).
For a positive integer 
, this harmonic mean
is
is the divisor function (the number of divisors
of 
).
, 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).
is a perfect number, 
is an integer.
excludes 
. Ore's conjecture implies that no odd
perfect numbers exist.
