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The harmonic number with = ⌊ ⌋ (red line) with its asymptotic limit + (blue line) where is the Euler–Mascheroni constant.. In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: [1] = + + + + = =.
The first few harmonic divisor numbers are 1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190 (sequence A001599 in the OEIS). Harmonic divisor numbers were introduced by Øystein Ore, who showed that every perfect number is a harmonic divisor number and conjectured that there are no odd harmonic divisor numbers other than 1.
6 is the 2nd superior highly composite number, [5] the 2nd colossally abundant number, [6] the 3rd triangular number, [7] the 4th highly composite number, [8] a pronic number, [9] a congruent number, [10] a harmonic divisor number, [11] and a semiprime. [12] 6 is also the first Granville number, or -perfect number.
The divergence of the harmonic series was first proven in 1350 by Nicole Oresme. [2] [4] Oresme's work, and the contemporaneous work of Richard Swineshead on a different series, marked the first appearance of infinite series other than the geometric series in mathematics. [5] However, this achievement fell into obscurity. [6]
The harmonic numbers are a fundamental sequence in number theory and analysis, known for their logarithmic growth. This result leverages the fact that the sum of the inverses of integers (i.e., harmonic numbers) can be closely approximated by the natural logarithm function, plus a constant, especially when extended over large intervals.
In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the fundamental frequency of a periodic signal. The fundamental frequency is also called the 1st harmonic ; the other harmonics are known as higher harmonics .
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In number theory, the harmonic numbers are the sums of the inverses of integers, forming the harmonic series. Harmonic number may also refer to: Harmonic, a periodic wave with a frequency that is an integral multiple of the frequency of another wave; Harmonic divisor numbers, also called Ore numbers or Ore's harmonic numbers, positive integers ...