q41=1151
【∑[k=1,41.q≠q+2]qk】=18485
2*1151^2=2649602
1/41=0.0243902439
1-0.0243902439=0.9756097561
2649602^0.9756097561=1847217
T(1847217)<18485
ln1847217^2=208.2015454142
1847217/18485=99.9305923722
99.9305923722/208.2015454142=0.4799704641
1/0.4799704641=2.0834615352

q42=1229
【∑[k=1,42.q≠q+2]qk】=19714
2*1229^2=3020882
1/42=0.0238095238
1-0.0238095238=0.9761904762
3020882^0.9761904762=2117605
T(2117605)<19714
ln2117605^2=212.1624215938
2117605/19714=107.4163031348
107.4163031348/212.1624215938=0.5062927842
1/0.5062927842=1.9751417188

q43=1277
【∑[k=1,43.q≠q+2]qk】=20991
2*1277^2=3261458
1/43=0.0232558140
1-0.0232558140=0.9767441860
3261458^0.9767441860=2301109
T(2301109)<20991
ln2301109^2=214.5903221472
2301109/20991=109.6236005907
109.6236005907/214.5903221472=0.5108506269
1/0.5108506269=1.9575193752

q44=1289
【∑[k=1,44.q≠q+2]qk】=22280
2*1289^2=3323042
1/44=0.0227272727
1-0.0227272727=0.9772727273
3323042^0.9772727273=2362214
T(2362214)<22280
ln2362214^2=215.3588497788
2362214/22280=106.0239676840
106.0239676840/215.3588497788=0.4923130291
1/0.4923130291=2.0312279808

q45=1301
【∑[k=1,45.q≠q+2]qk】=23581
2*1301^2=3385202
1/45=0.0222222222
1-0.0222222222=0.9777777778
3385202^0.9777777778=2423722
T(2423722)<23581
ln2423722^2=216.1139596973
2423722/23581=102.7828336373
102.7828336373/216.1139596973=0.4755955320
1/0.4755955320=2.1026269860

q46=1319
【∑[k=1,46.q≠q+2]qk】=24900
2*1319^2=3479522
1/46=0.0217391304
1-0.0217391304=0.9782608696
3479522^0.9782608696=2507914
T(2507914)<24900
ln2507914^2=217.1191018968
2507914/24900=100.7194377510
100.7194377510/217.1191018968=0.4638902652
1/0.4638902652=2.1556822271

q47=1427
【∑[k=1,47.q≠q+2]qk】=26327
2*1427^2=4072658
1/47=0.0212765957
1-0.0212765957=0.9787234043
4072658^0.9787234043=2946065
T(2946065)<26327
ln2946065^2=221.8902481529
2946065/26327=111.9027994075
111.9027994075/221.8902481529=0.5043159866
1/0.5043159866=1.9828838002

q48=1451
【∑[k=1,48.q≠q+2]qk】=27778
2*1451^2=4210802
1/48=0.0208333333
1-0.0208333333=0.9791666667
4210802^0.9791666667=3064484
T(3064484)<27778
ln3064484^2=223.0658673070
3064484/27778=110.3205414357
110.3205414357/223.0658673070=0.4945648690
1/0.4945648690=2.0219794463

q49=1481
【∑[k=1,49.q≠q+2]qk】=29259
2*1481^2=4386722
1/49=0.0204081633
1-0.0204081633=0.9795918367
4386722^0.9795918367=3210601
T(3210601)<29259
ln3210601^2=224.4593862776
3210601/29259=109.7303735603
109.7303735603/224.4593862776=0.4888651590
1/0.4888651590=2.0455538334

q50=1487
【∑[k=1,50.q≠q+2]qk】=30746
2*1487^2=4422338
1/50=0.02
1-0.02=0.98
4422338^0.98=3256409
T(3256409)<30746
ln3256409^=224.8840833085
3256409/30746=105.9132570090
105.9132570090/224.8840833085=0.4709682226
1/0.4709682226=2.1232855042

q100=3821
【∑[k=1,100.q≠q+2]qk】=163992
2*3821^2=29200082
1/100=0.01
1-0.01=0.99
29200082^0.99=24588398
T(24588398)<163992
ln24588398^2=289.6050152780
24588398/163992=149.9365700766
149.9365700766/289.6050152780=0.5177278091
1/0.5177278091=1.9315168751

q200=9629
【∑[k=1,200.q≠q+2]qk】=814052
2*9629^2=185435282
1/200=0.005
1-0.005=0.995
185435282^0.995=168597608
T(168597608)<814052
ln168597608^2=358.8382119112
168597608/814052=207.1091379912
207.1091379912/358.8382119112=0.5771657842
1/0.5771657842=1.7326044394

q300=17207
【∑[k=1,300.q≠q+2]qk】=2141494
2*17207^2=592161698
1/300=0.0033333333
1-0.0033333333=0.9966666667
592161698^0.9966666667=553603524
T(553603524)<2141494
ln553603524^2=405.2957863802
553603524/2141494=258.5127597836
258.5127597836/405.2957863802=0.6378372746
1/0.6378372746=1.5677979946

q400=23741
【∑[k=1,400.q≠q+2]qk】=4189920
2*23741^2=1127270162
1/400=0.0025
1-0.0025=0.9975
1127270162^0.9975=1070034898
T(1070034898)<4189920
ln1070034898^2=432.2638971257
1070034898/4189920=255.3831333295
255.3831333295/432.2638971257=0.5908037544
1/0.5908037544=1.6926094199

q500=32411
【∑[k=1,500.q≠q+2]qk】=7036532
2*32411^2=2100945842
1/500=0.002
1-0.002=0.998
2100945842^0.998=2012658203
T(2012658203)<7036532
ln2012658203^2=458.9330253615
2012658203/7036532=286.0298514950
286.0298514950/458.9330253615=0.6232496589
1/0.6232496589=1.6044934573

q600=41141
【∑[k=1,600.q≠q+2]qk】=10664032
2*41141^2=3385163762
1/600=0.0016666667
1-0.0016666667=0.9983333333
3385163762^0.9983333333=3263600944
T(3263600944)<10664032
ln3263600944^2=479.8770860713
3263600944/10664032=306.0381799304
306.0381799304/479.8770860713=0.6377428488
1/0.6377428488=1.5680301267

q700=49667
【∑[k=1,700.q≠q+2]qk】=15219834
2*49667^2=4933621778
1/700=0.0014285714
1-0.0014285714=0.9985714286
4933621778^0.9985714286=4778795797
T(4778795797)<15219834
ln4778795797^2=496.7306248089
4778795797/15219834=313.9847515420
313.9847515420/496.7306248089=0.6321026646
1/0.6321026646=1.5820214911

q800=58907
【∑[k=1,800.q≠q+2]qk】=20649236
2*58907^2=6940069298
1/800=0.00125
1-0.00125=0.99875
6940069298^0.99875=6746244892
T(6746244892)<20649236
ln6746244892^2=512.2188249199
6746244892/20649236=326.7067552523
326.7067552523/512.2188249199=0.6378265291
1/0.6378265291=1.5678244074

q900=69497
【∑[k=1,900.q≠q+2]qk】=27085306
2*69497^2=9659666018
1/900=0.0011111111
1-0.0011111111=0.9988888889
9659666018^0.9988888889=9416027295
T(9416027295)<27085306
ln9416027295^2=527.4224167860
9416027295/27085306=347.6433788490
347.6433788490/527.4224167860=0.6591365247
1/0.6591365247=1.5171363785

q1000=79559
【∑[k=1,1000.q≠q+2]qk】=34354616
2*79559^2=12659268962
1/1000=0.001
1-0.001=0.999
12659268962^0.999=12368192000
T(12368192000)<34354616
ln12368192000^2=540.0229488486
12368192000/34354616=360.0154343160
360.0154343160/540.0229488486=0.6666669168
1/0.6666669168=1.4999994372

q2000=182027
【∑[k=1,2000.q≠q+2]qk】=163443155
2*182027^2=66267657458
1/2000=0.0005
1-0.0005=0.9995
66267657458^0.9995=65447184452
T(65447184452)<163443155
ln65447184452^2=620.2345839769
65447184452/163443155=400.4278089957
400.4278089957/620.2345839769=0.6456070321
1/0.6456070321=1.5489298447

素数下界


p1=2
1+∑[k=1,1]pk=3
2^2=4
1/1=1
4^2=16
π(16)=6>3
ln16=2.7725887222
16/3=5.3333333333
5.3333333333/2.7725887222=1.9235933879
1/1.9235933879=0.5198603854

p2=3
1+∑[k=1,2]pk=6
3^2=9
1/2=0.5
9^1.5=27
π(27)=9>6
ln27=3.2958368660
27/6=4.5
4.5/3.2958368660=1.3653588399
1/1.3653588399=0.7324081925

p3=5
1+∑[k=1,3]pk=11
5^2=25
1/3=0.3333333333
25^1.3333333333=73
π(73)=21>11
ln73=4.2904594411
73/11=6.6363636364
6.6363636364/4.2904594411=1.5467722577
1/1.5467722577=0.6465075870

p4=7
1+∑[k=1,4]pk=18
7^2=49
1/4=0.25
49^1.25=129
π(129)=31>18
ln129=4.8598124044
129/18=7.1666666667
7.1666666667/4.8598124044=1.4746796934
1/1.4746796934=0.6781133588

p5=11
1+∑[k=1,5]pk=29
11^2=121
1/5=0.2
121^1.2=315
π(315)=65>29
ln315=5.7525726388
315/29=10.8620689655
10.8620689655/5.7525726388=1.8882106577
1/1.8882106577=0.5296019255

p6=13
1+∑[k=1,6]pk=42
13^2=169
1/6=0.1666666667
169^1.1666666667=397
π(397)=78>42
ln397=5.9839362807
397/42=9.4523809524
9.4523809524/5.9839362807=1.5796259367
1/1.5796259367=0.6330612690

p7=17
1+∑[k=1,7]pk=59
17^2=289
1/7=0.1428571429
289^1.1428571429=649
π(649)=118>59
ln649=6.4754327167
649/59=11
11/6.4754327167=1.6987281748
1/1.6987281748=0.5886757015

p8=19
1+∑[k=1,8]pk=78
19^2=361
1/8=0.1250000000
361^1.1250000000=753
π(753)=133>78
ln753=6.6240652278
753/78=9.6538461538
9.6538461538/6.6240652278=1.4573899595
1/1.4573899595=0.6861581511

p9=23
1+∑[k=1,9]pk=101
23^2=529
1/9=0.1111111111
529^1.1111111111=1061
π(1061)=178>101
ln1061=6.9669671386
1061/101=10.5049504950
10.5049504950/6.9669671386=1.5078225986
1/1.5078225986=0.6632079934

p10=29
1+∑[k=1,10]pk=130
29^2=841
1/10=0.1
841^1.1=1649
π(1649)=259>130
ln1649=7.4079243226
1649/130=12.6846153846
12.6846153846/7.4079243226=1.7123035863
1/1.7123035863=0.5840085882

@珞蘅 2016-05-14 11:36:11
数学工具也不是完善了之后才能实际运用的，而是在实际运用中得到完善和证明。
-----------------------------
这话不错！

p11=31
1+∑[k=1,11]pk=161
31^2=961
1/11=0.0909090909
961^1.0909090909=1794
π(1794)=278>161
ln1794=7.4922030426
1794/161=11.1428571429
11.1428571429/7.4922030426=1.4872604332
1/1.4872604332=0.6723771961

p12=37
1+∑[k=1,12]pk=198
37^2=1369
1/12=0.0833333333
1369^1.0833333333=2499
π(2499)=367>198
ln2499=7.8236459308
2499/198=12.6212121212
12.6212121212/7.8236459308=1.6132136133
1/1.6132136133=0.6198807100

p13=41
1+∑[k=1,13]pk=239
41^2=1681
1/13=0.0769230769
1681^1.0769230769=2976
π(2976)=429>239
ln2976=7.9983353960
2976/239=12.4518828452
12.4518828452/7.9983353960=1.5568092895
1/1.5568092895=0.6423394354

p14=43
1+∑[k=1,14]pk=282
43^2=1849
1/14=0.0714285714
1849^1.0714285714=3164
π(3164)=447>282
ln3164=8.0595923289
3164/282=11.2198581560
11.2198581560/8.0595923289=1.3921123672
1/1.3921123672=0.7183328182

p15=47
1+∑[k=1,15]pk=329
47^2=2209
1/15=0.0666666667
2209^1.0666666667=3690
π(3690)=514>329
ln3690=8.2133817370
3690/329=11.2158054711
11.2158054711/8.2133817370=1.3655526834
1/1.3655526834=0.7323042254

p16=53
1+∑[k=1,16]pk=382
53^2=2809
1/16=0.0625
2809^1.0625=4614
π(4614)=623>382
ln4614=8.4368504387
4614/382=12.0785340314
12.0785340314/8.4368504387=1.4316401742
1/1.4316401742=0.6984995378

p17=59
1+∑[k=1,17]pk=441
59^2=3481
1/17=0.0588235294
3481^1.0588235294=5623
π(5623)=739>441
ln5623=8.6346206083
5623/441=12.7505668934
12.7505668934/8.6346206083=1.4766794596
1/1.4766794596=0.6771950361

p18=61
1+∑[k=1,18]pk=502
61^2=3721
1/18=0.0555555556
3721^1.0555555556=5875
π(5875)=773>502
ln5875=8.6784613390
5875/502=11.7031872510
11.7031872510/8.6784613390=1.3485325098
1/1.3485325098=0.7415468242

p19=67
1+∑[k=1,19]pk=569
67^2=4489
1/19=0.0526315789
4489^1.0526315789=6988
π(6988)=898>569
ln6988=8.8519496713
6988/569=12.2811950791
12.2811950791/8.8519496713=1.3874000119
1/1.3874000119=0.7207726621

p20=71
1+∑[k=1,20]pk=640
71^2=5041
1/20=0.05
5041^1.05=7720
π(7720)=979>640
ln7720=8.9515696430
7720/640=12.0625000000
12.0625000000/8.9515696430=1.3475290347
1/1.3475290347=0.7420990378

p21=73
1+∑[k=1,21]pk=713
73^2=5329
1/21=0.0476190476
5329^1.0476190476=8018
π(8018)=1010>713
ln8018=8.9894442932
8018/713=11.2454417952
11.2454417952/8.9894442932=1.2509607300
1/1.2509607300=0.7993856050

p22=79
1+∑[k=1,22]pk=792
79^2=6241
1/22=0.0454545455
6241^1.0454545455=9284
π(9284)=1150>792
ln9284=9.1360477674
9284/792=11.7222222222
11.7222222222/9.1360477674=1.2830736573
1/1.2830736573=0.7793784825

p23=83
1+∑[k=1,23]pk=875
83^2=6889
1/23=0.0434782609
6889^1.0434782609=10116
π(10116)=1242>875
ln10116=9.2218736078
10116/875=11.5611428571
11.5611428571/9.2218736078=1.2536652907
1/1.2536652907=0.7976610722

p24=89
1+∑[k=1,24]pk=964
89^2=7921
1/24=0.0416666667
7921^1.0416666667=11514
π(11514)=1388>964
ln11514=9.3513189652
11514/964=11.9439834025
11.9439834025/9.3513189652=1.2772512035
1/1.2772512035=0.7829313429

p25=97
1+∑[k=1,25]pk=1061
97^2=9409
1/25=0.04
9409^1.04=13567
π(13567)=1605>1061
ln13567=9.5153956525
13567/1061=12.7869934025
12.7869934025/9.5153956525=1.3438215151
1/1.3438215151=0.7441464426

p26=101
1+∑[k=1,26]pk=1162
101^2=10201
1/26=0.0384615385
10201^1.0384615385=14548
π(14548)=1703>1162
ln14548=9.5852088061
14548/1162=12.5197934596
12.5197934596/9.5852088061=1.3061576136
1/1.3061576136=0.7656043877

p27=103
1+∑[k=1,27]pk=1265
103^2=10609
1/27=0.0370370370
10609^1.0370370370=14954
π(14954)=1751
ln14954=9.6127341016
14954/1265=11.8213438735
11.8213438735/9.6127341016=1.2297587501
1/1.2297587501=0.8131676233

p28=107
1+∑[k=1,28]pk=1372
107^2=11449
1/28=0.0357142857
11449^1.0357142857=15985
π(15985)=1861>1372
ln15985=9.6794060615
15985/1372=11.6508746356
11.6508746356/9.6794060615=1.2036766059
1/1.2036766059=0.8307879335

p29=109
1+∑[k=1,29]pk=1481
109^2=11881
1/29=0.0344827586
11881^1.0344827586=16419
π(16419)=1902>1481
ln16419=9.7061944798
16419/1481=11.0864280891
11.0864280891/9.7061944798=1.1422013140
1/1.1422013140=0.8755024073

p30=113
1+∑[k=1,30]pk=1594
113^2=12769
1/30=0.0333333333
12769^1.0333333333=17499
π(17499)=2014>1594
ln17499=9.7698990154
17499/1594=10.9780426600
10.9780426600/9.7698990154=1.1236597884
1/1.1236597884=0.8899490845

p31=127
1+∑[k=1,31]pk=1721
127^2=16129
1/31=0.0322580645
16129^1.0322580645=22046
π(22046)=2470>1721
ln22046=10.0008864585
22046/1721=12.8099941894
12.8099941894/10.0008864585=1.2808858737
1/1.2808858737=0.7807096796

p32=131
1+∑[k=1,32]pk=1852
131^2=17161
1/32=0.03125
17161^1.03125=23274
π(23274)=2594>1852
ln23274=10.0550921367
23274/1852=12.5669546436
12.5669546436/10.0550921367=1.2498099941
1/1.2498099941=0.8001216223

p33=137
1+∑[k=1,33]pk=1989
137^2=18769
1/33=0.0303030303
18769^1.0303030303=25289
π(25289)=2788>1989
ln25289=10.1381247976
25289/1989=12.7144293615
12.7144293615/10.1381247976=1.2541204232
1/1.2541204232=0.7973715933

p34=139
1+∑[k=1,34]pk=2128
139^2=19321
1/34=0.0294117647
19321^1.0294117647=25827
π(25827)=2842>2128
ln25827=10.1591757353
25827/2128=12.1367481203
12.1367481203/10.1591757353=1.1946587436
1/1.1946587436=0.8370591228

p35=149
1+∑[k=1,35]pk=2277
149^2=22201
1/35=0.0285714286
22201^1.0285714286=29549
π(29549)=3207>2277
ln29549=10.2938051813
29549/2277=12.9771629337
12.9771629337/10.2938051813=1.2606769513
1/1.2606769513=0.7932246235

p36=151
1+∑[k=1,36]pk=2428
151^2=22801
1/36=0.0277777778
22801^1.0277777778=30130
π(30130)=3258>2428
ln30130=10.3132766321
30130/2428=12.4093904448
12.4093904448/10.3132766321=1.2032442150
1/1.2032442150=0.8310864806

p37=157
1+∑[k=1,37]pk=2585
157^2=24649
1/37=0.0270270270
24649^1.0270270270=32396
π(32396)=3476>2585
ln32396=10.3857902374
32396/2585=12.5323017408
12.5323017408/10.3857902374=1.2066777255
1/1.2066777255=0.8287216867

p38=163
1+∑[k=1,38]pk=2748
163^2=26569
1/38=0.0263157895
26569^1.0263157895=34738
π(34738)=3709>2748
ln34738=10.4555894676
34738/2748=12.6411935953
12.6411935953/10.4555894676=1.2090369113
1/1.2090369113=0.8271046075

p39=167
1+∑[k=1,39]pk=2915
167^2=27889
1/39=0.0256410256
27889^1.0256410256=36259
π(36259)=3847>2915
ln36259=10.4984429053
36259/2915=12.4387650086
12.4387650086/10.4984429053=1.1848199891
1/1.1848199891=0.8440100684

p40=173
1+∑[k=1,40]pk=3088
173^2=29929
1/40=0.025
29929^1.025=38725
π(38725)=4081>3088
ln38725=10.5642406653
38725/3088=12.5404792746
12.5404792746/10.5642406653=1.1870686850
1/1.1870686850=0.8424112376

p41=179
1+∑[k=1,41]pk=3267
179^2=32041
1/41=0.0243902439
32041^1.0243902439=41266
π(41266)=4321>3267
ln41266=10.6277941953
41266/3267=12.6311600857
12.6311600857/10.6277941953=1.1885025108
1/1.1885025108=0.8413949410

p42=181
1+∑[k=1,42]pk=3448
181^2=32761
1/42=0.0238095238
32761^1.0238095238=41962
π(41962)=4388>3448
ln41962=10.6445197258
41962/3448=12.1699535963
12.1699535963/10.6445197258=1.1433069701
1/1.1433069701=0.8746557365

p43=191
1+∑[k=1,43]pk=3639
191^2=36481
1/43=0.0232558140
36481^1.0232558140=46575
π(46575)=4813>3639
ln46575=10.7488191955
46575/3639=12.7988458368
12.7988458368/10.7488191955=1.1907211019
1/1.1907211019=0.8398272260

p44=193
1+∑[k=1,44]pk=3832
193^2=37249
1/44=0.0227272727
37249^1.0227272727=47315
π(47315)=4879>3832
ln47315=10.7645826489
47315/3832=12.3473382046
12.3473382046/10.7645826489=1.1470336201
1/1.1470336201=0.8718140275

p45=197
1+∑[k=1,45]pk=4029
197^2=38809
1/45=0.0222222222
38809^1.0222222222=49080
π(49080)=5044>4029
ln49080=10.8012068988
49080/4029=12.1816827997
12.1816827997/10.8012068988=1.1278075602
1/1.1278075602=0.8866760920

p46=199
1+∑[k=1,46]pk=4228
199^2=39601
1/46=0.0217391304
39601^1.0217391304=49848
π(49848)=5119>4228
ln49848=10.8167336542
49848/4228=11.7899716178
11.7899716178/10.8167336542=1.0899752175
1/1.0899752175=0.9174520521

p47=211
1+∑[k=1,47]pk=4439
211^2=44521
1/47=0.0212765957
44521^1.0212765957=55907
π(55907)=5675>4439
ln55907=10.9314448749
55907/4439=12.5945032665
12.5945032665/10.9314448749=1.1521352768
1/1.1521352768=0.8679536337

p48=223
1+∑[k=1,48]pk=4662
223^2=49729
1/48=0.0208333333
49729^1.0208333333=62295
π(62295)=6257>4662
ln62295=11.0396364447
62295/4662=13.3622908623
13.3622908623/11.0396364447=1.2103922923
1/1.2103922923=0.8261784269

p49=227
1+∑[k=1,49]pk=4889
227^2=51529
1/49=0.0204081633
51529^1.0204081633=64300
π(64300)=6437>4889
ln64300=11.0713149102
64300/4889=13.1519738188
13.1519738188/11.0713149102=1.1879324114
1/1.1879324114=0.8417987340

p50=229
1+∑[k=1,50]pk=5118
229^2=52441
1/50=0.02
52441^1.02=65172
π(65172)=6512>5118
ln65172=11.0847852078
65172/5118=12.7338804220
12.7338804220/11.0847852078=1.1487710572
1/1.1487710572=0.8704954688

@强国之心不死 2016-05-16 23:51:24
说实话，我不太相信你的证明。我没有仔细看，所以也不能说你的肯定不对。
mark一下，有空哪天拜读一下。
-----------------------------
看过就会知道它的正确性和完整性。谢谢朋友回复！

p100=541
1+∑[k=1,100]pk=24134
541^2=292681
1/100=0.01
292681^1.01=331939
π(331939)=28561>24134
ln331939=12.7127064961
331939/24134=13.7539985083
13.7539985083/12.7127064961=1.0819095456
1/1.0819095456=0.9242916878

p200=1223
1+∑[k=1,200]pk=111588
1223^2=1495729
1/200=0.005
1495729^1.005=1605932
π(1605932)=
ln1605932=14.2892148314
1605932/111588=14.3916191705
14.3916191705/14.2892148314=1.0071665477
1/1.0071665477=0.9928844463

p300=1987
1+∑[k=1,300]pk=271062
1987^2=3948169
1/300=0.0033333333
3948169^1.0033333333=4153208
π(4153208)=
ln4153208=15.2393916056
4153208/271062=15.3219853760
15.3219853760/15.2393916056=1.0054197551
1/1.0054197551=0.9946094603

p400=2741
1+∑[k=1,400]pk=507826
2741^2=7513081
1/400=0.0025
7513081^1.0025=7816415
π(7816415)=
ln7816415=15.8717365675
7816415/507826=15.3919157349
15.3919157349/15.8717365675=0.9697688510
1/0.9697688510=1.0311735616

素数上界


p1=2
1+∑[k=1,1]pk=3
2^2=4
1/1=1
1-1=0
4^0=1
π(1)=0<3

p2=3
1+∑[k=1,2]pk=6
3^2=9
1/2=0.5
1-0.5=0.5
9^0.5=3
π(3)=2<6
ln3=1.0986122887
3/6=0.5
0.5/1.0986122887=0.4551196133
1/0.4551196133=2.1972245774

p3=5
1+∑[k=1,3]pk=11
5^2=25
1/3=0.3333333333
1-0.3333333333=0.6666666667
25^0.6666666667=9
π(9)=4<11
ln9=2.1972245773
9/11=0.8181818182
0.8181818182/2.1972245773=0.3723705927
1/0.3723705927=2.6854967057

p4=7
1+∑[k=1,4]pk=18
7^2=49
1/4=0.25
1-0.25=0.75
49^0.75=19
π(19)=8<18
ln19=2.9444389792
19/18=1.0555555556
1.0555555556/2.9444389792=0.3584912315
1/0.3584912315=2.7894685061

p5=11
1+∑[k=1,5]pk=29
11^2=121
1/5=0.2
1-0.2=0.8
121^0.8=47
π(47)=15<29
ln47=3.8501476017
47/29=1.6206896552
1.6206896552/3.8501476017=0.4209422139
1/0.4209422139=2.3756229881

p6=13
1+∑[k=1,6]pk=42
13^2=169
1/6=0.1666666667
1-0.1666666667=0.8333333333
169^0.8333333333=72
π(72)=20<42
ln72=4.2766661190
72/42=1.7142857143
1.7142857143/4.2766661190=0.4008462823
1/0.4008462823=2.4947219025

p7=17
1+∑[k=1,7]pk=59
17^2=289
1/7=0.1428571429
1-0.1428571429=0.8571428571
289^0.8571428571=129
π(129)=31<59
ln129=4.8598124044
129/59=2.1864406780
2.1864406780/4.8598124044=

p8=19
1+∑[k=1,8]pk=78
19^2=361
1/8=0.1250000000
1-0.1250000000=0.8750000000
361^0.8750000000=173
π(173)=40<78
ln173=5.1532915945
173/78=2.2179487179
2.2179487179/5.1532915945=0.4303945696
1/0.4303945696=2.3234493895

p9=23
1+∑[k=1,9]pk=101
23^2=529
1/9=0.1111111111
1-0.1111111111=0.8888888889
529^0.8888888889=264
π(264)=56<101
ln264=5.5759491031
264/101=2.6138613861
2.6138613861/5.5759491031=0.4687742549
1/0.4687742549=2.1332229523

p10=29
1+∑[k=1,10]pk=130
29^2=841
1/10=0.1
1-0.1=0.9
841^0.9=429
π(429)=82<130
ln429=6.0614569189
429/130=3.3
3.3/6.0614569189=0.5444235675
1/0.5444235675=1.8368051269

p11=31
1+∑[k=1,11]pk=161
31^2=961
1/11=0.0909090909
1-0.0909090909=0.9090909091
961^0.9090909091=515
π(515)=97<161
ln515=6.2441669007
515/161=3.1987577640
3.1987577640/6.2441669007=0.5122793505
1/0.5122793505=1.9520599435

p12=37
1+∑[k=1,12]pk=198
37^2=1369
1/12=0.0833333333
1-0.0833333333=0.9166666667
1369^0.9166666667=750
π(750)=132<198
ln750=6.6200732065
750/198=3.7878787879
3.7878787879/6.6200732065=0.5721808007
1/0.5721808007=1.7476993265

p13=41
1+∑[k=1,13]pk=239
41^2=1681
1/13=0.0769230769
1-0.0769230769=0.9230769231
1681^0.9230769231=950
π(950)=161<239
ln950=6.8564619846
950/239=3.9748953975
3.9748953975/6.8564619846=0.5797298091
1/0.5797298091=1.7249414888

p14=43
1+∑[k=1,14]pk=282
43^2=1849
1/14=0.0714285714
1-0.0714285714=0.9285714286
1849^0.9285714286=1081
π(1081)=180<282
ln1081=6.9856418176
1081/282=3.8333333333
3.8333333333/6.9856418176=0.5487446155
1/0.5487446155=1.8223413438

p15=47
1+∑[k=1,15]pk=329
47^2=2209
1/15=0.0666666667
1-0.0666666667=0.9333333333
2209^0.9333333333=1323
π(1323)=216<329
ln1323=7.1876571641
1323/329=4.0212765957
4.0212765957/7.1876571641=0.5594697276
1/0.5594697276=1.7874068080

p16=53
1+∑[k=1,16]pk=382
53^2=2809
1/16=0.0625
1-0.0625=0.9375
2809^0.9375=1711
π(1711)=265<382
ln1711=7.4448332739
1711/382=4.4790575916
4.4790575916/7.4448332739=0.6016330288
1/0.6016330288=1.6621427883

p17=59
1+∑[k=1,17]pk=441
59^2=3481
1/17=0.0588235294
1-0.0588235294=0.9411764706
3481^0.9411764706=2155
π(2154)=325<441
ln2154=7.6750818577
2154/441=4.8843537415
4.8843537415/7.6750818577=0.6363910942
1/0.6363910942=1.5713607703

p18=61
1+∑[k=1,18]pk=502
61^2=3721
1/18=0.0555555556
1-0.0555555556=0.9444444444
3721^0.9444444444=2357
π(2357)=349<502
ln2357=7.7651449029
2357/502=4.6952191235
4.6952191235/7.7651449029=0.6046531240
1/0.6046531240=1.6538407895

p19=67
1+∑[k=1,19]pk=569
67^2=4489
1/19=0.0526315789
1-0.0526315789=0.9473684211
4489^0.9473684211=2884
π(2884)=417<569
ln2884=7.9669334984
2884/569=5.0685413005
5.0685413005/7.9669334984=0.6361972648
1/0.6361972648=1.5718395148


p20=71
1+∑[k=1,20]pk=640
71^2=5041
1/20=0.05
1-0.05=0.95
5041^0.95=3292
π(3292)=462<640
ln3292=8.0992505618
3292/640=5.1437500000
5.1437500000/8.0992505618=0.6350896247
1/0.6350896247=1.5745809113