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Harmonic number
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{{Short description|Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n}} {{other uses}} {{Use American English|date = March 2019}} [[Image:HarmonicNumbers.svg|right|thumb|400px|The harmonic number <math>H_n</math> with <math>n=\lfloor x \rfloor</math> (red line) with its asymptotic limit <math>\gamma+\ln(x)</math> (blue line) where <math>\gamma</math> is the [[Euler–Mascheroni constant]].]] In [[mathematics]], the {{mvar|n}}-th '''harmonic number''' is the sum of the [[Multiplicative inverse|reciprocals]] of the first {{mvar|n}} [[natural number]]s:<ref>{{Cite book |last=Knuth |first=Donald |title=The Art of Computer Programming |publisher=Addison-Wesley |year=1997 |isbn=0-201-89683-4 |edition=3rd |pages=75–79 |language=en}}</ref> <math display="block">H_n= 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n} =\sum_{k=1}^n \frac{1}{k}.</math> Starting from {{math|1=''n'' = 1}}, the sequence of harmonic numbers begins: <math display="block">1, \frac{3}{2}, \frac{11}{6}, \frac{25}{12}, \frac{137}{60}, \dots</math> Harmonic numbers are related to the [[harmonic mean]] in that the {{mvar|n}}-th harmonic number is also {{mvar|n}} times the reciprocal of the harmonic mean of the first {{mvar|n}} positive integers. Harmonic numbers have been studied since antiquity and are important in various branches of [[number theory]]. They are sometimes loosely termed [[harmonic series (mathematics)|harmonic series]], are closely related to the [[Riemann zeta function]], and appear in the expressions of various [[special function]]s. The harmonic numbers roughly approximate the [[natural logarithm|natural logarithm function]]<ref name=ConwayGuy/>{{rp|143}} and thus the associated [[harmonic series (mathematics)|harmonic series]] grows without limit, albeit slowly. In 1737, [[Leonhard Euler]] used the [[Divergence of the sum of the reciprocals of the primes|divergence of the harmonic series]] to provide a new proof of the [[Euclid's theorem|infinity of prime numbers]]. His work was extended into the [[complex plane]] by [[Bernhard Riemann]] in 1859, leading directly to the celebrated [[Riemann hypothesis]] about the [[Prime number theorem|distribution of prime numbers]]. When the value of a large quantity of items has a [[Zipf's law]] distribution, the total value of the {{mvar|n}} most-valuable items is proportional to the {{mvar|n}}-th harmonic number. This leads to a variety of surprising conclusions regarding the [[long tail]] and [[Andrew Odlyzko#Network value|the theory of network value]]. The [[Bertrand's postulate|Bertrand-Chebyshev theorem]] implies that, except for the case {{math|1=''n'' = 1}}, the harmonic numbers are never integers.<ref name = 'ConcreteMath'>{{Cite book | first1 = Ronald L. | last1 = Graham | first2 = Donald E. | last2 = Knuth | first3 = Oren | last3 = Patashnik | title = Concrete Mathematics | year = 1994 | publisher = Addison-Wesley | title-link = Concrete Mathematics }}</ref> {| class="wikitable infobox collapsible collapsed" style="line-height:0.8;text-align:left;white-space:nowrap;" |+ The first 40 harmonic numbers ! rowspan="2" style="padding-top:1em;"|''n'' !! colspan="4"|Harmonic number, ''H<sub>n</sub>'' |- ! colspan="2"|expressed as a fraction !! decimal !! relative size |- | style="text-align:right;"|1 || style="text-align:center;" colspan="2"|1 || {{bartable|1||20}} |- | style="text-align:right;"|2 || style="border-right:none;padding-right:0;text-align:right;"|3 || style="border-left:none;padding-left:0;"|/2 || {{bartable|1.5||20}} |- | style="text-align:right;"|3 || style="border-right:none;padding-right:0;text-align:right;"|11 || style="border-left:none;padding-left:0;"|/6 || ~{{bartable|1.83333||20}} |- | style="text-align:right;"|4 || style="border-right:none;padding-right:0;text-align:right;"|25 || style="border-left:none;padding-left:0;"|/12 || ~{{bartable|2.08333||20}} |- | style="text-align:right;"|5 || style="border-right:none;padding-right:0;text-align:right;"|137 || style="border-left:none;padding-left:0;"|/60 || ~{{bartable|2.28333||20}} |- | style="text-align:right;"|6 || style="border-right:none;padding-right:0;text-align:right;"|49 || style="border-left:none;padding-left:0;"|/20 || {{bartable|2.45||20}} |- | style="text-align:right;"|7 || style="border-right:none;padding-right:0;text-align:right;"|363 || style="border-left:none;padding-left:0;"|/140 || ~{{bartable|2.59286||20}} |- | style="text-align:right;"|8 || style="border-right:none;padding-right:0;text-align:right;"|761 || style="border-left:none;padding-left:0;"|/280 || ~{{bartable|2.71786||20}} |- | style="text-align:right;"|9 || style="border-right:none;padding-right:0;text-align:right;"|7 129 || style="border-left:none;padding-left:0;"|/2 520 || ~{{bartable|2.82897||20}} |- | style="text-align:right;"|10 || style="border-right:none;padding-right:0;text-align:right;"|7 381 || style="border-left:none;padding-left:0;"|/2 520 || ~{{bartable|2.92897||20}} |- | style="text-align:right;"|11 || style="border-right:none;padding-right:0;text-align:right;"|83 711 || style="border-left:none;padding-left:0;"|/27 720 || ~{{bartable|3.01988||20}} |- | style="text-align:right;"|12 || style="border-right:none;padding-right:0;text-align:right;"|86 021 || style="border-left:none;padding-left:0;"|/27 720 || ~{{bartable|3.10321||20}} |- | style="text-align:right;"|13 || style="border-right:none;padding-right:0;text-align:right;"|1 145 993 || style="border-left:none;padding-left:0;"|/360 360 || ~{{bartable|3.18013||20}} |- | style="text-align:right;"|14 || style="border-right:none;padding-right:0;text-align:right;"|1 171 733 || style="border-left:none;padding-left:0;"|/360 360 || ~{{bartable|3.25156||20}} |- | style="text-align:right;"|15 || style="border-right:none;padding-right:0;text-align:right;"|1 195 757 || style="border-left:none;padding-left:0;"|/360 360 || ~{{bartable|3.31823||20}} |- | style="text-align:right;"|16 || style="border-right:none;padding-right:0;text-align:right;"|2 436 559 || style="border-left:none;padding-left:0;"|/720 720 || ~{{bartable|3.38073||20}} |- | style="text-align:right;"|17 || style="border-right:none;padding-right:0;text-align:right;"|42 142 223 || style="border-left:none;padding-left:0;"|/12 252 240 || ~{{bartable|3.43955||20}} |- | style="text-align:right;"|18 || style="border-right:none;padding-right:0;text-align:right;"|14 274 301 || style="border-left:none;padding-left:0;"|/4 084 080 || ~{{bartable|3.49511||20}} |- | style="text-align:right;"|19 || style="border-right:none;padding-right:0;text-align:right;"|275 295 799 || style="border-left:none;padding-left:0;"|/77 597 520 || ~{{bartable|3.54774||20}} |- | style="text-align:right;"|20 || style="border-right:none;padding-right:0;text-align:right;"|55 835 135 || style="border-left:none;padding-left:0;"|/15 519 504 || ~{{bartable|3.59774||20}} |- | style="text-align:right;"|21 || style="border-right:none;padding-right:0;text-align:right;"|18 858 053 || style="border-left:none;padding-left:0;"|/5 173 168 || ~{{bartable|3.64536||20}} |- | style="text-align:right;"|22 || style="border-right:none;padding-right:0;text-align:right;"|19 093 197 || style="border-left:none;padding-left:0;"|/5 173 168 || ~{{bartable|3.69081||20}} |- | style="text-align:right;"|23 || style="border-right:none;padding-right:0;text-align:right;"|444 316 699 || style="border-left:none;padding-left:0;"|/118 982 864 || ~{{bartable|3.73429||20}} |- | style="text-align:right;"|24 || style="border-right:none;padding-right:0;text-align:right;font-size:96%;"|1 347 822 955 || style="border-left:none;padding-left:0;font-size:96%;"|/356 948 592 || ~{{bartable|3.77596||20}} |- | style="text-align:right;"|25 || style="border-right:none;padding-right:0;text-align:right;font-size:87%;"|34 052 522 467 || style="border-left:none;padding-left:0;font-size:87%;"|/8 923 714 800 || ~{{bartable|3.81596||20}} |- | style="text-align:right;"|26 || style="border-right:none;padding-right:0;text-align:right;font-size:87%;"|34 395 742 267 || style="border-left:none;padding-left:0;font-size:87%;"|/8 923 714 800 || ~{{bartable|3.85442||20}} |- | style="text-align:right;"|27 || style="border-right:none;padding-right:0;text-align:right;font-size:80%;"|312 536 252 003 || style="border-left:none;padding-left:0;font-size:80%;"|/80 313 433 200 || ~{{bartable|3.89146||20}} |- | style="text-align:right;"|28 || style="border-right:none;padding-right:0;text-align:right;font-size:80%;"|315 404 588 903 || style="border-left:none;padding-left:0;font-size:80%;"|/80 313 433 200 || ~{{bartable|3.92717||20}} |- | style="text-align:right;"|29 || style="border-right:none;padding-right:0;text-align:right;font-size:73%;"|9 227 046 511 387 || style="border-left:none;padding-left:0;font-size:73%;"|/2 329 089 562 800 || ~{{bartable|3.96165||20}} |- | style="text-align:right;"|30 || style="border-right:none;padding-right:0;text-align:right;font-size:73%;"|9 304 682 830 147 || style="border-left:none;padding-left:0;font-size:73%;"|/2 329 089 562 800 || ~{{bartable|3.99499||20}} |- | style="text-align:right;"|31 || style="border-right:none;padding-right:0;text-align:right;font-size:64%;"|290 774 257 297 357 || style="border-left:none;padding-left:0;font-size:64%;"|/72 201 776 446 800 || ~{{bartable|4.02725||20}} |- | style="text-align:right;"|32 || style="border-right:none;padding-right:0;text-align:right;font-size:64%;"|586 061 125 622 639 || style="border-left:none;padding-left:0;font-size:64%;"|/144 403 552 893 600 || ~{{bartable|4.05850||20}} |- | style="text-align:right;"|33 || style="border-right:none;padding-right:0;text-align:right;font-size:68%;"|53 676 090 078 349 || style="border-left:none;padding-left:0;font-size:68%;"|/13 127 595 717 600 || ~{{bartable|4.08880||20}} |- | style="text-align:right;"|34 || style="border-right:none;padding-right:0;text-align:right;font-size:68%;"|54 062 195 834 749 || style="border-left:none;padding-left:0;font-size:68%;"|/13 127 595 717 600 || ~{{bartable|4.11821||20}} |- | style="text-align:right;"|35 || style="border-right:none;padding-right:0;text-align:right;font-size:68%;"|54 437 269 998 109 || style="border-left:none;padding-left:0;font-size:68%;"|/13 127 595 717 600 || ~{{bartable|4.14678||20}} |- | style="text-align:right;"|36 || style="border-right:none;padding-right:0;text-align:right;font-size:68%;"|54 801 925 434 709 || style="border-left:none;padding-left:0;font-size:68%;"|/13 127 595 717 600 || ~{{bartable|4.17456||20}} |- | style="text-align:right;"|37 || style="border-right:none;padding-right:0;text-align:right;font-size:60%;"|2 040 798 836 801 833 || style="border-left:none;padding-left:0;font-size:60%;"|/485 721 041 551 200 || ~{{bartable|4.20159||20}} |- | style="text-align:right;"|38 || style="border-right:none;padding-right:0;text-align:right;font-size:60%;"|2 053 580 969 474 233 || style="border-left:none;padding-left:0;font-size:60%;"|/485 721 041 551 200 || ~{{bartable|4.22790||20}} |- | style="text-align:right;"|39 || style="border-right:none;padding-right:0;text-align:right;font-size:60%;"|2 066 035 355 155 033 || style="border-left:none;padding-left:0;font-size:60%;"|/485 721 041 551 200 || ~{{bartable|4.25354||20}} |- | style="text-align:right;"|40 || style="border-right:none;padding-right:0;text-align:right;font-size:60%;"|2 078 178 381 193 813 || style="border-left:none;padding-left:0;font-size:60%;"|/485 721 041 551 200 || ~{{bartable|4.27854||20}} |} <!-- Python script to generate n from 2 to 40: import fractions numerator = 1; denominator = 1 for i in range(2, 40 + 1): numerator = numerator * i + denominator; denominator *= i; gcd = fractions.gcd(numerator, denominator); numerator /= gcd; denominator /= gcd decimal = ('{}' if (i < 3 or i == 6) else '{:.5f}').format(float(numerator) / denominator); exact = '' if (i < 3 or i == 6) else '~' numerator_length = len(str(numerator)); size = '' if (numerator_length <= 9) else 'font-size:{:d}%;'.format(960 / numerator_length) print('|-\n| style="text-align:right;"|{} || style="border-right:none;padding-right:0;text-align:right;{}"|{:,} || style="border-left:none;padding-left:0;{}"|/{:,} || {}{{{{bartable|{}||20}}}}'. format(i, size, numerator, size, denominator, exact, decimal).replace(',', ' ')) -->
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