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, ... (Sloane's 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.

Sloane, N. J. A. Sequences A099377 and A099378 in "The On-Line Encyclopedia of Integer Sequences."

CITE THIS AS:

Weisstein, Eric W. "Ore's Conjecture." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/OresConjecture.html