Harmonic Divisor Numbers and Perfect Numbers
For any integer M, as Ore observed, the product of the harmonic mean and arithmetic mean of its divisors equals M itself; as is obvious from the definition. Therefore, M is harmonic, with harmonic mean of divisors k, if and only if the average of its divisors is the product of M with a unit fraction 1/k.
Ore showed that every perfect number is harmonic. To see this, observe that the sum of the divisors of a perfect number M is exactly 2M; therefore, the average of the divisors is M(2/τ(M)), where τ(M) denotes the number of divisors of M. For any M, τ(M) is odd if and only if M is a square number, for otherwise each divisor d of M can be paired with a different divisor M/d. But, no perfect number can be a square: this follows from the known form of even perfect numbers and from the fact that odd perfect numbers (if they exist) must have a factor of the form qα where α ≡ 1 (mod 4). Therefore, for a perfect number M, τ(M) is even and the average of the divisors is the product of M with the unit fraction 2/τ(M); thus, M is a harmonic divisor number.
Ore conjectured that no odd harmonic divisor numbers exist other than 1. If the conjecture is true, this would imply the nonexistence of odd perfect numbers.
Read more about this topic: Harmonic Divisor Number
Famous quotes containing the words harmonic, numbers and/or perfect:
“For decades child development experts have erroneously directed parents to sing with one voice, a unison chorus of values, politics, disciplinary and loving styles. But duets have greater harmonic possibilities and are more interesting to listen to, so long as cacophony or dissonance remains at acceptable levels.”
—Kyle D. Pruett (20th century)
“The principle of majority rule is the mildest form in which the force of numbers can be exercised. It is a pacific substitute for civil war in which the opposing armies are counted and the victory is awarded to the larger before any blood is shed. Except in the sacred tests of democracy and in the incantations of the orators, we hardly take the trouble to pretend that the rule of the majority is not at bottom a rule of force.”
—Walter Lippmann (1889–1974)
“Why is it no one ever sent me yet
One perfect limousine, do you suppose?
Ah no, it’s always just my luck to get
One perfect rose.”
—Dorothy Parker (1893–1967)