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Harmonic number

The first 40 harmonic numbers
n Harmonic number, Hn
expressed as a fraction decimal relative size
1 1 1 1
 
2 3 /2 1.5 1.5
 
3 11 /6 ~1.83333 1.83333
 
4 25 /12 ~2.08333 2.08333
 
5 137 /60 ~2.28333 2.28333
 
6 49 /20 2.45 2.45
 
7 363 /140 ~2.59286 2.59286
 
8 761 /280 ~2.71786 2.71786
 
9 7 129 /2 520 ~2.82897 2.82897
 
10 7 381 /2 520 ~2.92897 2.92897
 
11 83 711 /27 720 ~3.01988 3.01988
 
12 86 021 /27 720 ~3.10321 3.10321
 
13 1 145 993 /360 360 ~3.18013 3.18013
 
14 1 171 733 /360 360 ~3.25156 3.25156
 
15 1 195 757 /360 360 ~3.31823 3.31823
 
16 2 436 559 /720 720 ~3.38073 3.38073
 
17 42 142 223 /12 252 240 ~3.43955 3.43955
 
18 14 274 301 /4 084 080 ~3.49511 3.49511
 
19 275 295 799 /77 597 520 ~3.54774 3.54774
 
20 55 835 135 /15 519 504 ~3.59774 3.59774
 
21 18 858 053 /5 173 168 ~3.64536 3.64536
 
22 19 093 197 /5 173 168 ~3.69081 3.69081
 
23 444 316 699 /118 982 864 ~3.73429 3.73429
 
24 1 347 822 955 /356 948 592 ~3.77596 3.77596
 
25 34 052 522 467 /8 923 714 800 ~3.81596 3.81596
 
26 34 395 742 267 /8 923 714 800 ~3.85442 3.85442
 
27 312 536 252 003 /80 313 433 200 ~3.89146 3.89146
 
28 315 404 588 903 /80 313 433 200 ~3.92717 3.92717
 
29 9 227 046 511 387 /2 329 089 562 800 ~3.96165 3.96165
 
30 9 304 682 830 147 /2 329 089 562 800 ~3.99499 3.99499
 
31 290 774 257 297 357 /72 201 776 446 800 ~4.02725 4.02725
 
32 586 061 125 622 639 /144 403 552 893 600 ~4.05850 4.0585
 
33 53 676 090 078 349 /13 127 595 717 600 ~4.08880 4.0888
 
34 54 062 195 834 749 /13 127 595 717 600 ~4.11821 4.11821
 
35 54 437 269 998 109 /13 127 595 717 600 ~4.14678 4.14678
 
36 54 801 925 434 709 /13 127 595 717 600 ~4.17456 4.17456
 
37 2 040 798 836 801 833 /485 721 041 551 200 ~4.20159 4.20159
 
38 2 053 580 969 474 233 /485 721 041 551 200 ~4.22790 4.2279
 
39 2 066 035 355 155 033 /485 721 041 551 200 ~4.25354 4.25354
 
40 2 078 178 381 193 813 /485 721 041 551 200 ~4.27854 4.27854
 

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:

Harmonic numbers are related to the harmonic mean in that the n-th harmonic number is also n times the reciprocal of the harmonic mean of the first n positive integers.

Harmonic numbers were studied in antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions.

The harmonic numbers roughly approximate the natural logarithm function and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the 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 distribution of prime numbers.

When the value of a large quantity of items has a Zipf's law distribution, the total value of the n most-valuable items is proportional to the n-th harmonic number. This leads to a variety of surprising conclusions in the Long Tail and the theory of network value.


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