Harmonic Number (disambiguation)

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Harmonic Number (disambiguation)

In Number Theory , the Harmonic Number s are the sums of the inverses of integers, forming the Harmonic Series .

In Engineering , the phrase 'harmonic number' is sometimes loosely used to refer to the ''n''th Harmonic of a Periodic Wave .

Philippe De Vitry , a 14th Century French Musicologist , defined a harmonic number as an integer that is not divisible by any prime other than 2 or 3. In 1342 , the Rabbi Gersonides , a contemporary of de Vitry, showed that there are only four pairs of such numbers that differ by one. The Abc Conjecture concerns a generalization of Gersonides results.

--- Ivars Peterson's MathTrek: '' Medieval Harmony ''

Ore's Harmonic Number s, defined by O. Ore in 1948 , are defined as those numbers for which $n au(n)/\sigma(n)$ is an integer, where $\sigma(n)$ is the Divisor Function , and $au(n)$ is the sum of divisors. Every Perfect Number satisfies Ore's condition.