Ore,s Conjecture

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.

REFERENCES:

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