第n个素数的极限系数k值
第n个素数不好求,一般用黎曼猜想公式反算,可以获得比较高的精度作为参考范围.这里不进行第n个素数的计算比赛,只对一个趋势做出预测,求出第n个素数的极限系数k值.
p(n)≈n×(lnn+lnlnn-k),这里n值和p(n)值都是已知的数据,只有k值未知,代人后
k=lnn+lnlnn-p(n)/n
n值 p(n)参考范围 计算的k值
4 10 -0.787071379
10 29 0.236617538
24 90 0.584322837
25 100 0.387908001
100 541 0.722349812
154 900 0.809598015
168 1000 0.805511381
500 3571 0.899510764
1000 7919 0.921400013
1117 9000 0.909640998
1229 10000 0.939318112
5000 48611 0.93708004
8713 90000 0.948434316
9592 100000 0.959124075
9592 100000 0.959124075
10000 104729 0.957767178
30000 350377 0.962732035
50000 611953 0.962094066
70000 882377 0.962864732
71274 900000 0.960576676
78498 999983 0.954084756
90000 1159523 0.958252788
100000 1299709 0.959305823
200000 2750159 0.957211231
300000 4256233 0.95870651
400000 5800079 0.956189157
500000 7368787 0.95910728
600000 8960453 0.958712825
602489 9000000 0.959219949
664579 10000000 0.955559154
700000 10570841 0.957269999
800000 12195257 0.957804142
900000 13834103 0.957060924
1000000 15485863 0.955439472
4310144 73473859 0.956066856
5216954 90000000 0.954714458
5761455 100000000 0.955107932
10000000 179424673 0.955570945
28944316 551987369 0.954016283
31255985 598608349 0.954181883
46009215 900000000 0.953469255
50847534 1000000000 0.953771771
62106578 1234567890 0.953423793
100000000 2038074743 0.953407301
111111111 2276855641 0.953517847
123456789 2543568463 0.953345453
399487944 8724310144 0.95293163
411523195 9000000000 0.952872778
449689285 9876543210 0.952957054
455052511 10000000000 0.952960865
498388617 11000000000 0.952836503
541555851 12000000000 0.952792438
551987369 12242181113 0.952827232
675522133 15125167433 0.95280727
1000000000 22801763489 0.952759371
1111111111 25458070277 0.952691408
1234567890 28423087477 0.952659797
2031255985 47824232591 0.952633346
3722428991 90000000000 0.952632842
4118054813 100000000000 0.952661953
4512105232 110000000000 0.952609693
4904759399 120000000000 0.952630576
8724310144 218695452037 0.952688936
9000000000 225898512559 0.952686974
9876543210 248857367251 0.952697866
10000000000 252097800623 0.952688406
11111111111 281329920307 0.952701468
12345678910 313945524931 0.952711638
15125167433 387829275881 0.952732747
24188924259 632071564697 0.952803332
33981987586 900000000000 0.952856627
37607912018 1000000000000 0.952882443
41220703418 1100000000000 0.952898557
44821651132 1200000000000 0.952913749
48411812843 1300000000000 0.952928255
51992079337 1400000000000 0.952943276
55563209929 1500000000000 0.952959823
59125832286 1600000000000 0.95296585
62680547806 1700000000000 0.952978131
66227858344 1800000000000 0.952997051
69768165361 1900000000000 0.953005511
73301896139 2000000000000 0.953018171
76829404177 2100000000000 0.953034877
80350940550 2200000000000 0.953031576
83866890515 2300000000000 0.953043799
87377470252 2400000000000 0.95305485
90000000000 2474799787573 0.95305936
90882915772 2500000000000 0.953063852
94383447111 2600000000000 0.95307122
97879280182 2700000000000 0.953081213
98765432109 2725369633577 0.953083106
100000000000 2760727302517 0.953090716
101370585277 2800000000000 0.953089006
104857545428 2900000000000 0.953097652
108340298703 3000000000000 0.953101171
111819048275 3100000000000 0.953113895
115293888086 3200000000000 0.953123695
118764942011 3300000000000 0.953129267
122232349643 3400000000000 0.953135088
125696244675 3500000000000 0.95314551
129156678096 3600000000000 0.95314744
132613795649 3700000000000 0.953151137
136067688083 3800000000000 0.953155352
139518461664 3900000000000 0.953163273
142966208126 4000000000000 0.953175813
146410910264 4100000000000 0.953173082
149852773964 4200000000000 0.953181021
153291810623 4300000000000 0.953187137
156728136258 4400000000000 0.95320009
160161747327 4500000000000 0.953206672
163592692077 4600000000000 0.953204523
167021151370 4700000000000 0.953215136
170447066694 4800000000000 0.953217788
173870573446 4900000000000 0.953226341
177291661649 5000000000000 0.953230035
180710431905 5100000000000 0.953237157
184126901158 5200000000000 0.953242428
187541119457 5300000000000 0.953246415
190953126302 5400000000000 0.953248309
194362990637 5500000000000 0.953252061
197770743547 5600000000000 0.953255983
201176428961 5700000000000 0.953260598
204580066580 5800000000000 0.953263125
207981734215 5900000000000 0.95326937
211381427039 6000000000000 0.95327346
214779167476 6100000000000 0.953273786
218175058254 6200000000000 0.953279837
221569102517 6300000000000 0.953287444
224961273873 6400000000000 0.953288951
228351698532 6500000000000 0.953296988
231740339225 6600000000000 0.953302771
235127223891 6700000000000 0.953306312
238512407609 6800000000000 0.953311095
241895851648 6900000000000 0.953309053
245277688804 7000000000000 0.953313329
248657916582 7100000000000 0.953320369
252036515078 7200000000000 0.953324842
255413505374 7300000000000 0.953326527
258788925595 7400000000000 0.953327242
259656817434 7425719040829 0.953329238
262162841215 7500000000000 0.953331879
265535241952 7600000000000 0.9533367
268906152117 7700000000000 0.95334203
272275559525 7800000000000 0.953344261
275643544693 7900000000000 0.953350004
279010070811 8000000000000 0.953353124
282375171661 8100000000000 0.953355251
285738881494 8200000000000 0.953358042
289101209285 8300000000000 0.953360526
292462199848 8400000000000 0.953365471
295821814466 8500000000000 0.953367157
299180106058 8600000000000 0.953369279
302537108851 8700000000000 0.953373572
305892791866 8800000000000 0.95337538
309247216816 8900000000000 0.95337922
312600354108 9000000000000 0.95338074
315952260435 9100000000000 0.953383952
319302948414 9200000000000 0.953388614
322652410741 9300000000000 0.953392682
326000647688 9400000000000 0.953394916
329347709593 9500000000000 0.953398697
332693561966 9600000000000 0.953399676
336038264769 9700000000000 0.953402105
339381817251 9800000000000 0.953404747
342724224549 9900000000000 0.953406947
346065536839 10000000000000 0.953411996
379418886337 11000000000000 0.953438621
511905471132 15000000000000 0.953527263
643217611048 19000000000000 0.953596753
675895909271 20000000000000 0.953610735
773621469799 23000000000000 0.953652834
886674661126 26486412297029 0.953696348
887114276903 26500000000000 0.953696396
900000000000 26898370231697 0.953701247
932911185326 27916701633581 0.953711916
967879555617 29000000000000 0.953724541
1000000000000 29996224275833 0.953735935
1000121668853 30000000000000 0.953735897
3204941750802 100000000000000 0.954126498
10000000000000 323780508946331 0.954537117
29844570422669 1000000000000000 0.954950335
100000000000000 3475385758524527 0.955423492
279238341033925 10000000000000000 0.955833997
547863431950008 20000000000000000 0.956106462
1000000000000000 37124508045065437 0.956350996
1075292778753150 40000000000000000 0.95638055
2623557157654233 100000000000000000 0.956744527
3893882469583623 150000000000000000 0.956905884
10000000000000000 394906913903735329 0.957291265
24739954287740860 1000000000000000000 0.957660554
48645161281738535 2000000000000000000 0.95793517
72254704797687083 3000000000000000000 0.95809533
95676260903887607 4000000000000000000 0.958208716
100000000000000000 4185296581467695669 0.958226556
118959989688273472 5000000000000000000 0.958296529
142135049412622144 6000000000000000000 0.958368158
165220513980969424 7000000000000000000 0.958428646
188229829247429504 8000000000000000000 0.958480986
211172979243258278 9000000000000000000 0.9585271
234057667276344607 10000000000000000000 0.958568315
460637655126005490 20000000000000000000 0.958838542
684559920583084690 30000000000000000000 0.958995813
706849401776940275 31000000000000000000 0.959008504
906790515105576571 40000000000000000000 0.959107022
1000000000000000000 44211790234832169331 0.959145642
1127779923790184543 50000000000000000000 0.959193055
2220819602560918840 100000000000000000000 0.959458995
3516585752930430595 160000000000000000000 0.959638155
3731397829842461168 170000000000000000000 0.959661194
3945938652811699917 180000000000000000000 0.9596829
4374267703076959271 200000000000000000000 0.959722871
4588082544160859769 210000000000000000000 0.959741366
5654086442526321042 260000000000000000000 0.959822191
5866733862193360875 270000000000000000000 0.959836454
6503696293016202398 300000000000000000000 0.95987623
6715709842660010419 310000000000000000000 0.9598886
6927578466326617308 320000000000000000000 0.959900571
8617821096373621600 400000000000000000000 0.959984581
10000000000000000000 465675465116607065549 0.96004168
10720710117789005897 500000000000000000000 0.960068355
12814731195053369962 600000000000000000000 0.960136628
21127269486018731928 1000000000000000000000 0.960327059
41644391885053857293 2000000000000000000000 0.960583407
82103246362658124007 4000000000000000000000 0.96083734
100000000000000000000 4892055594575155744537 0.960910633
201467286689315906290 10000000000000000000000 0.961169232
299751248358699805270 15000000000000000000000 0.961314699
397382840070993192736 20000000000000000000000 0.961417385
783964159847056303858 40000000000000000000000 0.961662996
1925320391606803968923 100000000000000000000000 0.96198374
18435599767349200867866 1000000000000000000000000 0.962769828
176846309399143769411680 10000000000000000000000000 0.96352742281009973617214866335650
1699246750872437141327603 100000000000000000000000000 0.96425691176903836741063255218860
从0.2一直上升,虽然区间有所反复,但总体趋势明显,就是趋向于1,即k值为1
所以无穷大时p(n)≈n×(lnn+lnlnn-1)成立
如果一直大下去,比1大,比2大,直至大到lnlnn,这时后两项一减为0,变成了p(n)≈n×lnn这可能吗?
我坚持无穷大时p(n)≈n×(lnn+lnlnn-1)
168*ln(384)=1000,
1229*ln(3417)=10000,
9592*ln(33703.41552)=10^5
455052512*ln(3498099876)≈10^10
p(n)≈n×(lnx),x值有一定的规律,但是还是逮不住它.
p(n)≈n×ln(n*log(n)),这么一算有点大了一点
p(n)≈n×ln(n*log(n)-loglog(n)),好一点了,大于10^10还是不理想
大于10^10,p(n)≈n×(lnn+lnlnn-0.955),精度又提升了
n*(ln(n*log(n))-1/(0.5*log(n)))
0.5这个系数到处都出现,只是不那么准确,它在0.4~0.6之间,由于无法预测,只好取0.5了
p(n)=n*(ln(n*logn)-k),log是以10为底的对数
举例子:9592*(ln(9592*log9592)-k)≈10^5
k值从一个负值往上升,-1.62...0.089....0.10...0.105...0.11...0.12...0.13
在0.11之间徘徊了好久,才升至0.12,升幅比较缓慢.
大胆预测一:p(n)=n*(ln(n*logn)-1)为极限,即无穷大时,k=1
p(n)=n*(ln(n*logn)-k),log是以10为底的对数
经过反复计算,这个k值到不了1,极限时,此值为0.16596755475204420019678695214200......
不能比这个再大了
页:
[1]