N* ∏(1-1p)是一个计算自然数N以下素数个数的上界公式

柳林 · 发表于 2018-2-14 15:18

N* ∏(1-1p)是一个计算自然数N以下素数个数的上界公式

请看下面计算结果：

密度          区间       近似值实际值误差率

33.3333333 21          7.00 8 -12.5000
26.6666667 84       22.40 23 -2.6087
22.8571429 189       43.20 42 2.8571
20.7792208 336    69.82 67 4.2062
19.1808192 525       100.70 99 1.7165
18.0525357 756       136.48 133 2.6144
17.1024022 1029 175.98 172 2.3161
16.3588195 1344 219.86 217 1.3191
15.7947223 1701 268.67 266 1.0031
15.2852151 2100 320.99 317 1.2585
14.8721012 2541 377.90 371 1.8599
14.5093670 3024 438.76 434 1.0975
14.1719399 3549 502.96 497 1.1996
13.8704093 4116 570.91 566 0.8668
13.6087034 4725 643.01 637 0.9437
13.3780474 5376 719.20 708 1.5825
13.1587352 6069 798.60 791 0.9613
12.9623362 6804 881.96 876 0.6801
12.7797680 7581 968.83 962 0.7104

柳林 · 发表于 2018-2-14 15:21

上帖有误，现在把正确的发出：

上界密度区间上界值实际值误差率
33.3333333 33 11.00 11 0.0000
26.6666667 132 35.20 32 10.0000
22.8571429 297 67.89 62 9.4931
20.7792208 528 109.71 99 10.8225
19.1808192 825 158.24 143 10.6586
18.0525357 1188 214.46 195 9.9816
17.1024022 1617 276.55 255 8.4494
16.3588195 2112 345.50 318 8.6473
15.7947223 2673 422.19 387 9.0938
15.2852151 3300 504.41 463 8.9443
14.8721012 3993 593.84 550 7.9715
14.5093670 4752 689.49 640 7.7321
14.1719399 5577 790.37 736 7.3871
13.8704093 6468 897.14 838 7.0570
13.6087034 7425 1010.45 941 7.3800
13.3780474 8448 1130.18 1057 6.9231
13.1587352 9537 1254.95 1180 6.3516

柳林 · 发表于 2018-2-14 15:22

12.7797680 11913 1522.45 1427 6.6891
12.6047027 13200 1663.82 1570 5.9758
12.4451495 14553 1811.14 1705 6.2254
12.2952080 15972 1963.79 1860 5.5801
12.1570596 17457 2122.26 2007 5.7428
12.0317290 19008 2286.99 2159 5.9283
11.9126030 20625 2456.97 2324 5.7218
11.7969467 22308 2631.66 2500 5.2665
11.6866948 24057 2811.47 2675 5.1016
11.5794775 25872 2995.84 2846 5.2650
11.4770042 27753 3185.21 3028 5.1920
11.3866341 29700 3381.83 3221 4.9932
11.2997132 31713 3583.48 3409 5.1182
11.2172336 33792 3790.53 3619 4.7396
11.1365340 35937 4002.14 3817 4.8503
11.0617922 38148 4219.85 4025 4.8411
10.9885353 40425 4442.12 4237 4.8411
10.9185446 42768 4669.64 4472 4.4196
10.8515597 45177 4902.41 4688 4.5736
10.7865803 47652 5140.02 4911 4.6634
10.7242301 50193 5382.81 5153 4.4598
10.6643182 52800 5630.76 5388 4.5056
10.6053993 55473 5883.13 5631 4.4776
10.5498737 58212 6141.29 5896 4.1603
10.4952111 61017 6403.86 6147 4.1787
10.4419359 63888 6671.14 6406 4.1390
10.3894639 66825 6942.76 6659 4.2613
10.3402247 69828 7220.37 6921 4.3256
10.2938560 72897 7503.91 7206 4.1342
10.2485086 76032 7792.15 7487 4.0757
10.2037553 79233 8084.74 7765 4.1177

柳林 · 发表于 2018-2-14 15:23

10.1599624 82500 8381.97 8065 3.9302
10.1174521 85833 8684.11 8351 3.9889
10.0754709 89232 8990.54 8643 4.0211
10.0353296 92697 9302.45 8954 3.8916
9.9962817 96228 9619.22 9270 3.7672
9.9582730 99825 9940.85 9577 3.7992
9.9212534 103488 10267.31 9890 3.8150
9.8846436 107217 10598.02 10216 3.7394
9.8489589 111012 10933.53 10536 3.7730
9.8139093 114873 11273.53 10863 3.7792
9.7792311 118800 11617.73 11198 3.7482
9.7458549 122793 11967.23 11546 3.6483
9.7141095 126852 12322.54 11892 3.6204
9.6828744 130977 12682.34 12242 3.5969
9.6519387 135168 13046.33 12589 3.6328
9.6214909 139425 13414.76 12960 3.5090
9.5924230 143748 13788.92 13324 3.4893
9.5639588 148137 14167.76 13682 3.5504
9.5363970 152592 14551.78 14072 3.4095
9.5090721 157113 14939.99 14444 3.4339
9.4821342 161700 15332.61 14824 3.4310
9.4557216 166353 15729.88 15200 3.4860
9.4299566 171072 16132.02 15591 3.4700
9.4046753 175857 16538.78 15977 3.5162
9.3798608 180708 16950.16 16398 3.3672
9.3553703 185625 17365.91 16799 3.3746
9.3313205 190608 17786.24 17212 3.3363
9.3078159 195657 18211.39 17625 3.3271
9.2846044 200772 18640.89 18042 3.3194
9.2619037 205953 19075.17 18458 3.3436
9.2397989 211200 19514.46 18908 3.2074
9.2178516 216513 19957.85 19323 3.2854
9.1964645 221892 20406.22 19768 3.2285
9.1752256 227337 20858.68 20206 3.2301
9.1543253 232848 21315.66 20663 3.1586
9.1336609 238425 21776.93 21096 3.2278
9.1133187 244068 22242.69 21550 3.2144
9.0933770 249777 22713.16 22025 3.1245
9.0736517 255552 23187.90 22487 3.1169
9.0540542 261393 23666.66 22942 3.1587
9.0346665 267300 24149.66 23416 3.1332
9.0158050 273273 24637.76 23909 3.0481
8.9972920 279312 25130.52 24380 3.0784
8.9789676 285417 25627.50 24871 3.0417
8.9609737 291588 26129.12 25356 3.0491
8.9431586 297825 26634.96 25832 3.1084
8.9255886 304128 27145.21 26334 3.0805
8.9084569 310497 27660.49 26838 3.0647
8.8914235 316932 28179.77 27352 3.0263
8.8749884 323433 28704.64 27879 2.9615
8.8587635 330000 29233.92 28404 2.9218
8.8428591 336633 29767.98 28924 2.9179
8.8271524 343332 30306.44 29447 2.9186
8.8116390 350097 30849.28 29984 2.8858
8.7962070 356928 31396.13 30514 2.8909
8.7809623 363825 31947.34 31048 2.8966
8.7660033 370788 32503.29 31580 2.9236
8.7512208 377817 33063.60 32130 2.9057
8.7366111 384912 33628.26 32676 2.9143
8.7220743 392073 34196.90 33232 2.9035
8.7077051 399300 34769.87 33806 2.8512
8.6935001 406593 35347.16 34349 2.9059
8.6794101 413952 35928.59 34910 2.9178
8.6653885 421377 36513.95 35493 2.8765
8.6516557 428868 37104.18 36068 2.8729
8.6381586 436425 37699.08 36676 2.7895
8.6247244 444048 38297.92 37270 2.7580
8.6113941 451737 38900.85 37851 2.7736
8.5982066 459492 39508.07 38429 2.8080
8.5851593 467313 40119.57 39005 2.8575
8.5721711 475200 40734.96 39632 2.7830
8.5594339 483153 41355.16 40249 2.7483
8.5467907 491172 41979.44 40856 2.7498
8.5342771 499257 42607.98 41479 2.7218
8.5219265 507408 43240.94 42098 2.7149

柳林 · 发表于 2018-2-14 15:31

古人云：一生二，二生三，三生万物。N=3*5n^2可以得到下界值，N=3*7n^2可以得到近似值值，

N=3*11n^2可以得到上界值。这是“道法自然”的最好证明。

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