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Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n
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
Harmonic_number
Positive integer whose divisors have a harmonic mean that is an integer
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
Harmonic_divisor_number
Topics referred to by the same term
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
Harmonic number (disambiguation)
Harmonic_number_(disambiguation)
Divergent sum of positive unit fractions
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions: ∑ i = 1 ∞ 1 i = 1 1 + 1 2 + 1 3 + 1 4 + 1 5 +
Harmonic_series_(mathematics)
Sequence of frequencies
The harmonic series (also overtone series) is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a fundamental
Harmonic_series_(music)
{\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
List of logarithmic identities
List_of_logarithmic_identities
Area of mathematical analysis
Harmonic analysis is an area of mathematical analysis that emerged from the study of harmonic functions, and especially their boundary behavior. The methods
Harmonic_analysis
Mathematical terminology
function Harmonic mean Harmonic mode Harmonic number Harmonic series Alternating harmonic series Harmonic tremor Spherical harmonics This set index article
Harmonic_(mathematics)
Lowest frequency of a periodic waveform, such as sound
first harmonic and the first partial. The numbering of the partials and harmonics is then usually the same; the second partial is the second harmonic, etc
Fundamental_frequency
Number equal to the sum of its proper divisors
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
Perfect_number
Non-sinusoidal waveform
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
Triangle_wave
Inverse of the average of the inverses of a set of numbers
In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is sometimes used for ratios and rates such as speeds, and is
Harmonic_mean
Number that is the numerator of the generalized harmonic number H_(n,2)
In mathematics, 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,
Wolstenholme_number
Natural number
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
616_(number)
Addition of several numbers or other values
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
Summation
Natural number
natural number after 279 and before 281. The denominator of the eighth harmonic number, 280 is an octagonal number. 280 is the smallest octagonal number that
280_(number)
Special mathematical functions defined on the surface of a sphere
fields. The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of orthogonal
Spherical_harmonics
Probability distribution
}}\ N<k~.\end{cases}}} where HN is a normalization constant: The Nth harmonic number: H N ≡ ∑ k = 1 N 1 k . {\displaystyle H_{N}\equiv \sum _{k=1}^{N}{\frac
Zipf's_law
Natural number
Square root of 2 −2 Colman, Samuel (1912). Coan, C. Arthur (ed.). Nature's Harmonic Unity: A Treatise on Its Relation to Proportional Form. New York and London:
2
Count of permutations by cycles
2}\right)+2H_{n-1,3}\right)\\\end{aligned}}} where Hn is the harmonic number H n = 1 1 + 1 2 + … + 1 n {\displaystyle H_{n}={\frac {1}{1}}+{\frac
Stirling numbers of the first kind
Stirling_numbers_of_the_first_kind
Rational number sequence
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
Bernoulli_number
String instrument technique
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
String_harmonic
Mathematical problem of placing fuel depots
base. The distance travelled on the last trip is the nth harmonic number, Hn. As the harmonic numbers are unbounded, it is possible to exceed any given
Jeep_problem
Difference between logarithm and harmonic series
Greek letter gamma (γ), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log: γ = lim n → ∞ (
Euler's_constant
Mathematical manifold theory
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
Hodge_theory
Measure of genetic diversity
_{i=1}^{n-1}{1 \over i}} is the ( n − 1 ) {\displaystyle (n-1)} th harmonic number. This estimate is based on coalescent theory. The Watterson estimator
Watterson_estimator
Discrete probability distribution
{1}{(i+q)^{s}}},} which may be thought of as a generalization of a harmonic number. In the formula, k {\displaystyle k} is the rank of the data, and q
Zipf–Mandelbrot_law
Topics referred to by the same term
hn, the Internet country code top-level domain (ccTLD) for Honduras Harmonic number, in mathematics Hellenic Navy, the navy of Greece Hospitalman, a US
HN
Functions in mathematics
mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R {\displaystyle
Harmonic_function
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
List_of_prime_numbers
Wave with frequency an integer multiple of the fundamental frequency
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
Harmonic
particular, H n = H n ( 1 ) {\displaystyle H_{n}=H_{n}^{(1)}} is the n-th harmonic number. The hyperharmonic numbers were discussed by J. H. Conway and R. K
Hyperharmonic_number
Quantum mechanical model
The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually
Quantum_harmonic_oscillator
Number theory theorem
H_{n}=1+{\tfrac {1}{2}}+{\tfrac {1}{3}}+\cdots +{\tfrac {1}{n}}} is the nth harmonic number. Generalizations of Lucas's theorem for higher prime powers pk are
Lucas's_theorem
Probability of shared birthdays
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
Birthday_problem
Mathematical subject
combinatorics is a field in the intersection of number theory, combinatorics, ergodic theory and harmonic analysis. Arithmetic combinatorics is about combinatorial
Arithmetic_combinatorics
Natural number
a Hexagonal number. 496 is also a harmonic divisor number. The group E8 has real dimension 496. The number 496 is a very important number in superstring
496_(number)
30 apart. 1638 = 2 × 32 × 7 × 13. It is a harmonic divisor number. 1639 = 11 × 149. It is a nonagonal number. 1646 = 2 × 823. There are 1646 graphs with
1000_(number)
Classical problem in combinatorics
where H ( n ) {\displaystyle H(n)} is the n {\displaystyle n} -th harmonic number: H ( n ) = ∑ k = 1 n 1 k ≤ ln n + 1 {\displaystyle H(n)=\sum _{k=1}^{n}{\frac
Set_cover_problem
Natural number
number, a harmonic divisor number, and a semiprime. 6 is also the first Granville number, or S {\displaystyle {\mathcal {S}}} -perfect number. A Golomb
6
Proton synchrotron in Russia
acceleration voltage (RF) frequency, frf 5.5-6.1 MHz HF multiplicity (harmonic number) 30 Betatron tune, νx = νy 9.75-9.85 intensity of the beam of protons
U-70_(synchrotron)
Natural number
natural number following 27 and preceding 29. 28 is a composite number, a perfect number, a harmonic divisor number, a centered nonagonal number, a hexagonal
28_(number)
Measurement of the harmonic distortion present in a signal
The total harmonic distortion (THD or THDi) is a measurement of the harmonic distortion present in a signal and is defined as the ratio of the sum of the
Total_harmonic_distortion
Integration by parts version of Abel's method for summation by parts
}{u^{2}}}\,du.} The left-hand side is the harmonic number H ⌊ x ⌋ {\displaystyle H_{\lfloor x\rfloor }} . Fix a complex number s {\displaystyle s} . If a n = 1
Abel's_summation_formula
Concept in mathematics
differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential
Harmonic_map
Extension of the factorial function
} where H ( m ) {\displaystyle H(m)} is the m {\displaystyle m} th harmonic number and γ {\displaystyle \gamma } is the Euler–Mascheroni constant. For
Gamma_function
Conjecture on zeros of the zeta function
(n)<H_{n}+\log(H_{n})e^{H_{n}}} for every natural number n > 1, where H n {\displaystyle H_{n}} is the nth harmonic number. The Riemann hypothesis is also true if
Riemann_hypothesis
Concept in combinatorics
{z^{k}}{k}}=H_{m}{\mbox{ for }}n\geq m} where Hm is the mth harmonic number. Hence the expected number of cycles of length at most m in a random permutation
Random_permutation_statistics
Vibration damping system in an engine
A harmonic damper is a device fitted to the free (accessory drive) end of the crankshaft of an internal combustion engine to counter torsional and resonance
Harmonic_damper
Mathematics problem
31183,} where H n {\displaystyle H_{n}} is the n {\displaystyle n} -th harmonic number. Therefore, using the cycle-following strategy the prisoners survive
100_prisoners_problem
Natural number
tuning for musical instruments because it provides a good approximation of harmonic intervals. January, March, May, July, August, October and December have
31_(number)
Problem in probability theory
H_{n}.\end{aligned}}} Here Hn is the n-th harmonic number. Using the asymptotics of the harmonic numbers, we obtain: E ( T ) = n ⋅ H n = n log
Coupon_collector's_problem
Natural number
ISBN 978-0-521-87818-0. Colman, Samuel (1912). Coan, C. Arthur (ed.). Nature's Harmonic Unity: A Treatise on Its Relation to Proportional Form. New York and London:
1
Aspect of music
effects created by distinct pitches or tones coinciding with one another; harmonic objects such as chords, textures and tonalities are identified, defined
Harmony
Topics referred to by the same term
note. Harmonic (mathematics), a number of concepts in mathematics Harmonic analysis, representing signals by superposition of basic waves Harmonic oscillator
Harmonic_(disambiguation)
Mathematical function
_{0}^{1}\left({\frac {1-t^{z}}{1-t}}\right)\,dt.} The integral is Euler's harmonic number H z {\displaystyle H_{z}} , so the previous formula may also be written
Digamma_function
hyperbolic space if the sum is less than 1. A harmonic divisor number is a positive integer whose divisors have a harmonic mean that is an integer. The first five
List_of_sums_of_reciprocals
Second-order partial differential equation
twice continuously differentiable solutions of Laplace's equation are the harmonic functions, which are important in multiple branches of physics, notably
Laplace's_equation
Physical system that responds to a restoring force proportional to displacement
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional
Harmonic_oscillator
Brass instrument
or sound production. Every pitch in a glissando must have the same harmonic number, and a tritone is the largest interval that can be performed as a glissando
Trombone
American technology company
Harmonic Inc. is an American technology company that develops and markets video routing, server, and storage products for companies that produce, process
Harmonic_Inc.
Branch of pure mathematics
than for large sieves. The study of the latter now includes ideas from harmonic and functional analysis. The Galois group of an extension L/K consists
Number_theory
In Riemannian geometry, a branch of mathematics, harmonic coordinates are a certain kind of coordinate chart on a smooth manifold, determined by a Riemannian
Harmonic_coordinates
Nonlinear optical process
Harmonic generation (HG, also called multiple harmonic generation) is a nonlinear optical process in which n {\displaystyle n} photons with the same frequency
Harmonic_generation
Arithmetic function related to the divisors of an integer
(n)<H_{n}+e^{H_{n}}\log(H_{n})} for every natural number n > 1, where H n {\displaystyle H_{n}} is the nth harmonic number, (Lagarias 2002). Divisor sum convolutions
Divisor_function
Quantum number denoting orbital angular momentum
ground state. The angular momentum quantum number, ℓ and the corresponding spherical harmonic govern the number of planar nodes going through the nucleus
Azimuthal_quantum_number
Natural number
hundred [and] seventy) is the natural number following 269 and preceding 271. 270 is a harmonic divisor number. 270 degrees is equal to three-fourths
270_(number)
Mechanical transmission system with flexing
Strain wave gearing (also known as harmonic gearing) is a type of mechanical gear system that uses a flexible spline with external teeth, which is deformed
Strain_wave_gearing
Natural number
[and] forty) is the natural number following 139 and preceding 141. 140 is an abundant number and a harmonic divisor number. It is the sum of the squares
140_(number)
Measurement device for radio equipment
reflected power. The approximation improves as crosstalk weakens and harmonic number increases. Over time, nonlinear high gain amplifiers have replaced
SWR_meter
Hash based data structure
n}{H_{k}}},} where H k {\displaystyle H_{k}} is the k {\displaystyle k} -th harmonic number. Since H k / log k → 1 {\displaystyle H_{k}/\log k\to 1} as k → ∞
Kademlia
Australian and American mathematician (born 1975)
contributions to partial differential equations, combinatorics, harmonic analysis, and additive number theory. He is a professor of mathematics at the University
Terence_Tao
The scale of harmonics is a musical scale based on the noded positions of the natural harmonics existing on a string.[citation needed] This musical scale
Scale_of_harmonics
Number of prime factors of a natural number
_{p^{k}||n}{H_{k}}} where H k {\displaystyle H_{k}} is the k {\displaystyle k} -th harmonic number and ε {\displaystyle \varepsilon } is the identity for the Dirichlet
Prime_omega_function
Function whose domain is the positive integers
nth harmonic number. Then σ ( n ) ≤ H n + e H n log H n {\displaystyle \sigma (n)\leq H_{n}+e^{H_{n}}\log H_{n}} is true for every natural number n
Arithmetic_function
Operation on formal power series
\\j\end{matrix}}\right\}_{\ast }}} , appears below. These weighted-harmonic-number expansions are almost identical to the known formulas for the Stirling
Generating function transformation
Generating_function_transformation
Multiple-winner electoral system
who are elected, that ballot contributes the r {\displaystyle r} -th harmonic number to that committee's vote total. In other words: H ( r ) = 1 + 1 2 +
Proportional_approval_voting
A harmonic spectrum is a spectrum containing only frequency components whose frequencies are whole number multiples of the fundamental frequency; such
Harmonic_spectrum
Equivalence of average-case and expected complexity
H_{k}=1+{\tfrac {1}{2}}+\cdots +{\tfrac {1}{k}}} is the k {\displaystyle k} th harmonic number. By renewal theory, the offline algorithm incurs n ( k + 1 ) H k +
Yao's_principle
Natural number
664 = harmonic divisor number 108,968 = number of signed trees with 11 nodes 109,376 = automorphic number 110,880 = 30th highly composite number 111,111
100,000
mathematicians to prove similar formulas. Let k {\displaystyle k} be a number field, A {\displaystyle \mathbb {A} } be its adele ring, k × {\displaystyle
Waldspurger_formula
Nonlinear optical process
Second-harmonic generation (SHG), also known as frequency doubling, is the lowest-order wave-wave nonlinear interaction that occurs in various systems
Second-harmonic_generation
Progression formed by taking the reciprocals of an arithmetic progression
In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression, which is
Harmonic progression (mathematics)
Harmonic_progression_(mathematics)
Natural number
because 13 is prime. The next prime number is 43, making both twin primes. It is a regular prime,a Ramanujan prime, a harmonic prime, a good prime, a Newman–Shanks–Williams
41_(number)
Sum of inverse squares of natural numbers
H_{n}^{(2)}:=\sum _{k=1}^{n}k^{-2}} denote a generalized second-order harmonic number, we can easily prove by induction that [ x 2 ] ∏ k = 1 n ( 1 − x 2
Basel_problem
Constants in the zeta function's Laurent series expansion
{B_{2N+2}\cdot H_{2N+1}}{2N+2}},\qquad 0<\theta <1,} where Hn is the nth harmonic number. More complicated series for Stieltjes constants are given in works
Stieltjes_constants
to use as few bins as possible, but minimizing the number of bins is an NP-hard problem. The harmonic bin-packing algorithms rely on partitioning the items
Harmonic_bin_packing
{\displaystyle k} th vertex from the root is the k {\displaystyle k} th harmonic number, from which it follows by linearity of expectation that the sum of
Random_recursive_tree
Analytic function in mathematics
{2}}}}\right)} There are yet more formulas in the article Harmonic number. There are a number of related zeta functions that can be considered to be generalizations
Riemann_zeta_function
exists with the price of stability in this game being at most the nth harmonic number in directed graphs. For undirected graphs, Anshelevich et al. presented
Price_of_stability
Mathematical objects more general than vectors
{\displaystyle l} is the number of indices, i.e., tensor rank; B. Tensor is symmetric with respect to indices; C. Tensor is harmonic, i.e., it is a solution
Harmonic_tensors
Difference in pitch between two notes
sounding tones, such as two adjacent pitches in a melody, and vertical or harmonic if it pertains to simultaneously sounding tones, such as in a chord. In
Interval_(music)
American mathematician
who works in the areas of analytic number theory, automorphic forms and representation theory, L-functions, harmonic analysis, and homogeneous dynamics
Alex_Kontorovich
Atomic model
often partially replaced by cubic harmonics for a number of reasons. These harmonics are usually named tesseral harmonics in the field of condensed matter
Cubic_harmonic
Probability distribution
H ( N , α − 1 ) {\displaystyle H(N,\alpha -1)} is the generalized harmonic number. This makes Zipf's probability density function derivable from Pareto's
Pareto_distribution
Categorizations of simultaneous or successive sounds
perception of harmonic partials of the sounds considered, to such an extent that the distinction really holds only in the case of harmonic sounds (i.e.
Consonance_and_dissonance
Number used for counting
natural-number results: subtracting a larger natural number from a smaller one results in a negative number and dividing one natural number by another
Natural_number
Formal power series
solve for closed-form solutions to P-recurrences involving generalized harmonic numbers. Other packages listed on this particular RISC site are targeted
Generating_function
Algorithm for anomaly detection
the testing data size, m is the size of the sample set and H is the harmonic number, which can be estimated by H ( i ) = l n ( i ) + γ {\displaystyle H(i)=ln(i)+\gamma
Isolation_forest
American mathematician
H_{n}+e^{H_{n}}\ln H_{n}} with equality only when n = 1. Here Hn is the nth harmonic number, the sum of the reciprocals of the first n {\displaystyle n} positive
Jeffrey_Lagarias
HARMONIC NUMBER
HARMONIC NUMBER
Girl/Female
English
Unity; concord; musically in tune. Harmonia was the mythological daughter of Aphrodite.
Girl/Female
American, Australian, British, Chinese, Christian, English, French, Greek, Latin
A State of Order or Agreement; A Beautiful Blending; Agreement; Concord; Musical Combination of Chords; Harmony; Joining
Girl/Female
American, British, English, Greek, Latin
A State of Order or Agreement; Unity; Concord; Musically in Tune; A Tuneful Sound
Girl/Female
Greek Latin
Daughter of Ares.
Boy/Male
Indian
Harmony
Boy/Male
Muslim
Harmony
Male
English
English surname transferred to forename use, from the German personal name Harman, HARMON means "bold/hardy man."
Girl/Female
English
Unity; concord; musically in tune. Harmonia was the mythological daughter of Aphrodite.
Girl/Female
Latin
Harmony.
Female
Greek
(ΑÏμονία) Greek name HARMONIA means "concord, harmony." In mythology, this is the name of the daughter of Ares and Aphrodite. Her Latin name is Concordia.
Girl/Female
Christian & English(British/American/Australian)
Harmony
Surname or Lastname
Irish (mainly County Louth)
Irish (mainly County Louth) : generally of English origin (see 1); but sometimes also used as a variant of Harman or Hardiman, i.e. an Anglicized form of Gaelic Ó hArgadáin (see Hargadon).English : variant spelling of Harman 1.
Boy/Male
American, Australian, British, Chinese, Christian, English, French, German, Greek, Hebrew
Man of the Army; Army Man; Noble; Name of a Place During Biblical Period; Hardy Man; Variant of Herman
Boy/Male
Welsh
Harmony.
Female
English
English name derived from the vocabulary word harmony, from Greek Harmonia, HARMONY means "concord, harmony."
Girl/Female
Latin American
Concord.
Girl/Female
Latin
Harmony.
Female
English
Variant spelling of English Harmony, HARMONIE means "concord, harmony."
Boy/Male
French American Hebrew
Girl/Female
American, Australian, British, Christian, English, French, Greek, Latin
A State of Order or Agreement; Unity; Concord; Harmony; Agreement
HARMONIC NUMBER
HARMONIC NUMBER
HARMONIC NUMBER
HARMONIC NUMBER
HARMONIC NUMBER
HARMONIC NUMBER
HARMONIC NUMBER