Ore's Conjecture

Define the harmonic mean of the divisors of n


where sigma_0(n) is the divisor function (the number of divisors of n).

For n=1, 2, ..., the values of H(n) 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 n is a perfect number, H(n) is an integer.

Ore conjectured that if n is odd, then H(n) is not an integer. This implies that no odd perfect numbers exist.

See also

Harmonic Divisor Number, Harmonic Mean, Odd Perfect Number, Perfect Number

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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--A Wolfram Web Resource.

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