Harmonic number (disambiguation)

From Wikipedia, the free encyclopedia

The term harmonic number has several different meanings:

• In number theory, the harmonic numbers 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 nth 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 numbers, defined by O. Ore in 1948, are defined as those numbers for which nτ(n) / σ(n) is an integer, where σ(n) is the divisor function, and τ(n) is the sum of divisors. Every perfect number satisfies Ore's condition.