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Date: 4-1-2021
1417
Date: 13-11-2019
773
Date: 2-2-2021
1381
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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."
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