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