In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: H n = 1 + 1 2 + 1 3 + ⋯ + 1 n = ∑ k = 1 n 1 k ...

mathematics, a harmonic divisor number or Ore number is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor...

Look up harmonic number in Wiktionary, the free dictionary. In number theory, the harmonic numbers are the sums of the inverses of integers, forming the...

In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: ∑ n = 1 ∞ 1 n = 1 + 1 2 + 1 3 + 1 4 + 1 5 + ⋯...

A harmonic series (also overtone series) is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a fundamental...

1st harmonic; the other harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also...

In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for...

function Harmonic mean Harmonic mode Harmonic number Harmonic series Alternating harmonic series Harmonic tremor Spherical harmonics This article includes...

{\displaystyle k} -th harmonic number, defined as H k = ∑ j = 1 k 1 j {\displaystyle H_{k}=\sum _{j=1}^{k}{\frac {1}{j}}} The harmonic numbers are a fundamental...

interchangeably with harmonic analysis. Harmonic analysis has become a vast subject with applications in areas as diverse as number theory, representation...

perfect number should exist. All perfect numbers are also harmonic divisor numbers, and it has been conjectured as well that there are no odd harmonic divisor...

odd harmonics. However, the higher harmonics roll off much faster than in a square wave (proportional to the inverse square of the harmonic number as opposed...

Playing a string harmonic (a flageolet) is a string instrument technique that uses the nodes of natural harmonics of a musical string to isolate overtones...

mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R , {\displaystyle...

4+36+576=616. The 616th harmonic number is the first to exceed seven. 666 is generally believed to have been the original Number of the Beast in the Book...

{163}{60}}=2+{\frac {43}{60}},\ } which is also five minus the fifth harmonic number. Every solvable configuration of the Fifteen puzzle can be solved in...

= 363 {\displaystyle 11\times 33=363} is the seventh numerator of harmonic number H 7 {\displaystyle H_{7}} , where specifically, the previous such numerators...

fields. The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of orthogonal...

In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. Equivalently...

collector's problem. It can be calculated by nHn, where Hn is the nth harmonic number. For 365 possible dates (the birthday problem), the answer is 2365...

Hk ≡ 0 (mod p) and Hk ≡ −ωp (mod p) for 1 ≤ k ≤ p−2, where Hk denotes the k-th harmonic number and ωp denotes the Wolstenholme quotient. 5, 13, 17, 23, 41, 67, 73...

Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log: γ = lim n → ∞ (...

(the nth harmonic number) ∑ i = 1 n 1 i k = H n k {\displaystyle \sum _{i=1}^{n}{\frac {1}{i^{k}}}=H_{n}^{k}\quad } (a generalized harmonic number) The following...

A Wolstenholme number is a number that is the numerator of the generalized harmonic number Hn,2. The first such numbers are 1, 5, 49, 205, 5269, 5369...

{1}{H_{N}}}\,{\frac {1}{k}}} where HN is a normalization constant, the Nth harmonic number: H N = ∑ k = 1 N 1 k   . {\displaystyle H_{N}=\sum _{k=1}^{N}{\frac...

A harmonic spectrum is a spectrum containing only frequency components whose frequencies are whole number multiples of the fundamental frequency; such...

k+1}\right\}.} A Bernoulli number is then introduced as an inclusion–exclusion sum of Worpitzky numbers weighted by the harmonic sequence 1, ⁠1/2⁠, ⁠1/3⁠...

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional...

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually...

vanishes under the Laplacian operator of the metric. Such forms are called harmonic. The theory was developed by Hodge in the 1930s to study algebraic geometry...